"House of wickedness" is not the only possibility for what Hosea means here. In both Joshua and Samuel, the name "Beth-aven" (בֵּית אָוֶן) is a place distinct from Bethel, but nearby it.
Joshua next sent men from Jericho to Ai, which is near Beth-aven and east of Bethel (Joshua 7:2)
Saul chose three thousand of Israel, of whom two thousand remained with him in Michmash and in the hill country of Bethel... They came up and encamped in Michmash, east of Beth-aven.
In Hosea, Beth-haven is the place where one of the golden calves was located (the other was in the northern town of Dan). It is not completely certain that he was using "aven" pejoratively. The word "aven" often came to be associated with evil, but its origins seem to be related to "proper" or working in vain. According to Stong's, "aven" is:
From an unused root perhaps meaning properly, to pant (hence, to exert
oneself, usually in vain; to come to naught); strictly nothingness;
also trouble. Vanity, wickedness; specifically an idol -- affliction,
evil, false, idol, iniquity, mischief, mourners(-ing), naught, sorrow,
unjust, unrighteous, vain, vanity, wicked(-ness)
Regarding Gilgal, it was an important center of Israelite worship in earlier days. Joshua, with the assistance of the priests, established a memorial of twelve stones there (Joshua 4:20). It was one of the several sanctuaries visited yearly by Samuel (1 Samuel 7:16). Saul's kingship was to be confirmed there (1 Samuel 11:14). However, in Hosea's day, people were required to go to the altar of Jerusalem to offer their sacrifices. Hosea denounces Gilgal and its memorial stones in dramatic terms:
In Gilead is falsehood, they have come to nothing;
in Gilgal they sacrifice bulls, But their altars are like heaps of stones
in the furrows of the field. (Hosea 12:12)
Conclusion: Beth-aven was originally a distinct location from Bethel. However, Hosea clearly speaks of the "calf of Beth-aven," so he equates the two places. Beth-aven may have been absorbed into Bethel in Hosea's time. If Hosea was engaging in wordplay with aven, then either "vanity" or "iniquity/evil" would be his meaning. Gilgal and Bethel were both accepted sanctuaries of Israelite worship in earlier days but were no longer approved and had fallen into corruption and/or idolatry. Hosea denounces both of them an warns the Israelites not to "prostitute themselves" by worshiping there.