Many say a man has spiritual headship over a wife and use Ephesians Ch 5 to prove this. There is a problem with this as it contradicts the teaching in 1 Pet 2:24-3:7.
We know The Bible does not contradict itself so one assumption is incorrect.
The probability is that Ephesians was not written by Paul but by someone using his name who had not been taught about headship by Jesus. Scholars think the following books were not written by Paul but by people using his name as a form of reverence: Ephesians, 1 and 2 Thessalonians, 1 and 2 Timothy and Titus. So one must be careful to have what The Bible teaches else where what is being used from these books.
Peter was an apostle and would have been taught by Jesus how Jesus viewed headship in marriage.
So we examine the teaching of Peter.
1Pe 2:25 For you had gone astray like sheep, and you have returned now to The Shepherd and The Caregiver of your souls.
We now turn to a common case in the early church. Two non-Christians marry and the wife becomes a Christian. So is she to submit totally to the man as she would have before marriage which means observing his beliefs and practices? Hers are now different and to do what the man does could be idolatry and sin.
What is she to do?
1Pe 3:1 In this way, also, you women, submit to your husbands, that you may win without labor those who do not obey the word by your beautiful way of life,
1Pe 3:2 When they see that you conduct yourselves in reverence and discretion.
1Pe 3:3 Do not be adorned with outward ornaments of your hair braids or of gold jewelry or fine clothing,
1Pe 3:4 But be adorned in the secret person of the heart and in a humble spirit which is incorrupted, an excellent ornament before God
. 1Pe 3:5 For in this way from the first, the holy women who hoped in God were adorning their souls and they submitted to their husbands.
1Pe 3:6 Just as Sarah was subject to Abraham and was calling him, "My lord", whose daughters you are by good works, when you are not shaken with any fear.
This submission is so that the husband can care for her and protect her. If she does not submit domestically he cannot do this properly and it will be to her own hurt.
If the man becomes a Christian he is to look after her domestically. He does not have to submit to her domestically but has stewardship of her according to God.
1Pe 3:7 And you men, in this way dwell with your wives by knowledge, and hold them in honor as weaker vessels, lest you be subverted in your prayers, because they also inherit the gift* of eternal life with you.
As Christ is the head of the family (both of them) Headship questions should not arise as if there is a problem they should ask Jesus what to do and not argue or try to reason it out. This is why we need to be led by The Holy Spirit in everything.
In the Garden God made eve a helpmeet - one who helps and is not subservient spiritually but only domestically. Adam had stewardship of Eve and The Garden but failed. He did not protect the tree and did not protect ever from eating its fruit. This is why it is said that Adam sinned.
The wife's responsibility is to obey Jesus and not the husband but out of obedience to Jesus will be to be a good wife and obey the husband domestically.