In a previous post 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)

In a previous post Lift chain complex from $\mathbb{F}_2$ to $\mathbb{Z}$ 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.)