Here are three quotes from Zwingli, Calvin and Beza (Geneva Bible Translation Notes) that teach that head covering was only a cultural tradition and not necessary if the culture changed.
ULRICH ZWINGLI
“This passage refers to either those teaching or listening to the Word of God. Paul wants men to uncover their heads in the public assembly. Let us discern whether they should follow this from general principal, or for a specific purpose. To be sure, Paul is not laying down everlasting and unchanging laws; instead he is giving instructions as to what is customary and honorable. Therefore this custom does not mandate that men must keep their heads completely bare during winter-time and frost, especially in cold regions or when the cold hurts the head. Naturally, people are free from certain external practices, but in such a way that the public custom is upheld. That which maintains public decency and is customary should not be defiantly disregarded.”
(Annotations on the Evangelical History of our Lord Jesus Christ, p. 473. Zurich, 1529)
JOHN CALVIN
“Let us observe that St. Paul has only taken exception to something that was not appropriate and fitting according to the usage of the land. For (as we have shown) we are not to take those countries and measure them by our custom(s). Yet there was such disorder, as we said, that the men had exchanged (places) with the women, which was intolerable. This is why St. Paul says that a man, when he executes the function of prophesying, or prays on behalf of all so that all may answer, “Amen,” must not have his head covered, as if he hid himself for shame.“
(Men, Women, and Order in the Church: Three Sermons by John Calvin, Presbyterian Heritage Publications, 1992, pp. 12, 24, 25).
A Facebook friend has also written about Calvin and head covering here. John Calvin – Head Covering History . Calvin seems to go back and forth.
THEODORE BEZA (GENEVA BIBLE TRANSLATION NOTES)
“1 Corinthians 11:4
(3) Every (b) man praying or prophesying, having [his] head covered, dishonoureth his head.
(3) By this he gathers that if men do either pray or preach in public assemblies having their heads covered (which was then a sign of subjection), they robbed themselves of their dignity, against God’s ordinance.
(b) It appears, that this was a political law serving only for the circumstance of the time that Paul lived in, by this reason, because in these our days for a man to speak bareheaded in an assembly is a sign of subjection.”
(The Geneva Bible Translation Notes [1599])
Beza appeared to emphasise the “political law” of men uncovering more than women covering.
There are a number of other similar quotes from the time of the Reformation to today by some surprising people.
As for God the Father’s authority, I did say:
To be honest, however, I would appreciate a more thorough explanation as to why the authority of God the Father is the fundamental thing on display in a woman’s head covering. I accept that the image of God includes the authority of the Father shown in the man’s authority over the woman (vv7, 10), but this is not explicit. Is there more you can tell me?