This arXiv paper on bicrossed products (knit products) gives an elementary introduction, classifies some examples, and shows $A_6$ cannot be written as such a product:

EDIT: Sorry, my comments on the content of that paper are based on an earlier version (v2).

Version 2 of "Factorization problems for finite groups", which discusses bicrossed products (knit products), gives an elementary introduction, classifies some examples, and shows that $A_6$ cannot be written as such a product.