In <a href="https://mathoverflow.net/questions/163346/lift-chain-complex-from-mathbbf-2-to-mathbbz">a previous post</a> the body of the question mentions that this is always possible : 

"Now, this can always be done (for example, by using the structure theorem for chain complexes over F2)."

What would be a constructive way of doing this?

(unlike the original post, I'm removing the other two restrictions)