In a previous post https://mathoverflow.net/questions/163346/lift-chain-complex-from-mathbbf-2-to-mathbbz the body of the question mentions that this (lifting a chain complex from $\mathbb Z/2\mathbb Z$ to $\mathbb Z$) is always possible : 

"Now, this can always be done (for example, by using the structure theorem for chain complexes over $\mathbb F_2$)."

What would be a constructive way of doing this?

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