If Christ is the head and the church is the bride and the husband is under Christ, this implies that the husband is responsible for covenant and doctrinal teachings and values of the house. I think the gospel is number one here first before anything else, but secondarily come the needs of the household. If the wife is to compliment the husband in the house, it would naturally vary from household to household depending on the need and the dependency… But primarily I would definitely believe that the gospel and doctrinal matters would fall under the responsibility of the man of the household.
The husband is also called to submit to the wife. The husband is accountable to Christ, but the wife has also been regenerated and saved by this same grace, and therefore the wife should be willing to keep the husband accountable. She is a helpmate, but before any demands of the household come, she should be a helpmate to keep him accountable before Christ.
Simply an observation, open to criticism