This is an excellent point. The third person singular is indicative of the change from the plural used in the verses preceding vs. 11. Beginning with vs. 12, the form changes to "a woman", not "women". Paul instructs Timothy on the manner of dress and conduct for women praying in public in vs. 9.
"8 I wish, therefore, that men pray in every place, lifting up kind hands, apart from anger and reasoning; 9 in like manner also the women, in becoming apparel, with modesty and sobriety to adorn themselves, not in braided hair, or gold, or pearls, or garments of great price,"
Many commentaries have been written on these verses because they are confusing, and seem out of sync with the first 10 verses, and with other scriptures Paul wrote concerning women praying and prophesying in the assemblies (1 Cor. 11:5), and commending Phebe who was a "ministrant of the assembly," and who "became a leader of many, and of myself" (Rom. 16:1-2).
Becoming a leader of many, including of Paul, would seem to indicate some form of teaching on the part of Phebe.
There were problems in the church at Ephesus which Paul identified in Ephesians and in his letters to Timothy.
"First, they have lost their spiritual bearings. They have wandered (v. 6)--an image of slow but steady movement away from some point. Perhaps in the beginning these teachers only drifted aimlessly. But as they hardened in their disbelief and became argumentative in their attempts to convince others of their views, their lives came to be characterized not by love but by controversy, impure hearts and ineffective consciences. They have wandered from the faith.
Second, they speak and teach foolishness (v. 6). Having left the faith and diverged from the standard of approved teaching, their doctrines and discussions are meaningless talk, devoid of truth (6:4-5). In choosing the word he does, Paul places their doctrine into the category of idolatry and paganism (compare Acts 14:15; 1 Pet 1:18).
Third, verse 7 reveals that they claim authority for their teaching. Teachers of the law, a title given to the rabbis (Lk 5:17; Acts 5:34), were regarded as the authoritative interpreters of Scripture. These enthusiasts were not interested in simply offering their ideas for consideration. Rather, they "taught" them as God's message and expected them to be received." Source: False Teachers
In the letter to the church at Ephesus, Christ told them in Rev. 2:4-5,
"4 Nevertheless I have this against you, that you have left your first love. 5 Remember therefore from where you have fallen; ..." (NKJV)
They had fallen away from their first love of Christ and His word, and were turning back to fables, rabbinical legalism, and idolatry.
Paul identified three men who were most likely the source of these false teachings: Hymaneus and Alexander (1 Tim. 1:20) and Philetus (2 Tim. 2:17). Paul's instructions to Timothy were to remain in Ephesus so that he "may charge some that they teach no other doctrine," (1 Tim. 1:3), and to
" Hold fast the pattern of sound words which you have heard from me, in faith and love which are in Christ Jesus." 2 Tim. 1:13. (NKJV)
So, Paul's concern was that the church at Ephesus not stray from their faith in Christ. 1 Tim. 6:20-21,
"20 O Timothy! Guard what was committed to your trust, avoiding the profane and idle babblings and contradictions of what is falsely called knowledge— 21 by professing it some have strayed concerning the faith..." (NKJV)
In 1 Tim. 2:12, The Greek word for "woman" is "gykaiki", is from Strong's 1135 "gune" and means "a woman, wife, or my lady". It was sometimes used for a specific woman or wife such as in John 4:42, and Acts 5:1. Other times it is used in the generic sense for all women as in 1 Cor. 7:27 and 1 Cor. 11:6.
So, the word in the third person singular cannot by itself be an indicator of whether verses 11-15 of 1 Tim. 2 refer to all women or just one at the church at Ephesus who may have been deceived by false teachers.
Nor can Paul's comment about being saved through child-bearing be a universal condition for salvation of women as it does not meet the conditions of Acts 2:38 and Mark 16:16 by which we all must be saved, nor does it address the condition of unmarried women with no children, nor of married women who are barren and may never have children.
So, the "they" third person plural pronoun in vs. 15 has no other antecedent but the third person singular "woman" of verse 14.
Because of the confusion and difficulty of these verses in 1. Tim chap.2, and because they seem to oppose previous scriptures which Paul taught in the letters to the Corinthians and Romans for the conduct of women in the assemblies and the work they had done with Paul, the admonition to Timothy seems to be centered around the false teachers and deceivers at Ephesus who were causing some women at Ephesus to be led astray.
One interesting commentary on the child-bearing in verse 15 is based on 1 Cor. 3:15, and was put forth by Henry Alford in 1863:
" Just as that man should be saved through, as passing through, fire which is his trial, his hindrance in his way, in spite of which he escapes — so she shall be saved, through, as passing through, her child-bearing, which is her trial, her curse, her (not means of salvation, but) hindrance in the way of it. (Alford, H. . Alford’s Greek Testament: an exegetical and critical commentary [Vol. 3, 320]. Bellingham, WA: Logos Bible Software.)" Source: here