@Jim you have great/ cybersynchronous timing to attack this problem. any reaction on this? and @Thomas what about you?

@Jim Interesting question, never noticed before that permutation groups are a nice example for understanding Sylow subgroups. You seem to be right, one column in the correct position seems to be always achievable. And Tobias' permutation sending $i$ to $n-i+1$ seems to show that you can't get more than one column right in general.

@vzn Hi vzn, Sir, hope you are well. yes I am aware of babai but can not get it as it uses Luks' work, If u remember , I had(have) a little knowledge about group theory, thus can't get Luks' which disable me to read babai's latest work. i have started Group theory, hope to get both Luks' and Babai soon.

@ThomasKlimpel I think , 1 column is not achievable, I missed a simple fact, sorry for wasting your time!

@Jim If you can prove that H cannot always permute one column to the correct position, then you shoudl write that as an answer. Or was this a typo, and you meant that more than one column is not achievable?

I think one column should be achievable, since H will always be transitive on more than half of the elements. No permutation will be able to complete move all elements from that domain away.