Can a man give a woman permission to lead a part of the worship service?

Answer:

You know the answer. Men never have been given the right to override God. To give permission to another to break a law of God is a sin. "Let your women keep silent in the churches, for they are not permitted to speak; but they are to be submissive, as the law also says" (I Corinthians 14:34). It isn't a popular rule, but it is the rule God gave.

What the song leader mistakenly believed is that as long as he gave permission, then the woman would not be usurping authority as Paul taught against in "And I do not permit a woman to teach or to have authority over a man, but to be in silence" (I Timothy 2:12). The King James Version says "usurp authority," but this translation is misleading in modern English. "A recent article (1984) by George Knight ("Authenteo in Reference to Women") thoroughly analyzes the occurrences of authenteo, and shows that in its earliest uses (First Century B.C. to Second Century A.D.) it means simply "to have authority." "The authority in view in the documents is understood to be a positive concept and is in no way regarded as having any overtones of misuse of position or power, i.e. 'to domineer'" (pp. 150-151). Even as late as Sixth Century A.D., this basic sense still predominated." [The Complete Biblical Library]. However, a second flaw is that just because he gives permission to have a woman over him, it does not imply the entire church gives permission. But the real point is that he cannot give permission where God has not.

What I don't know is why you did not stand up and rebuke this brother for his sin, and if not accepted by the congregation, walk out.