Since you wrote "2-torsion", I'm going to assume that $\mathbb{Z}_2$ refers to $\mathbb{Z}/2$ (rather than the $2$-adic integers, which I think is the preferred interpretation of the notation $\mathbb{Z}_2$ in those parts of algebraic topology close to algebra nowadays).

In that case, the short exact sequence $0\to \mathbb{Z} \to \mathbb{Z}\to \mathbb{Z}/2\to 0$ induces a long exact sequence on cohomology groups, exhibiting $H^*(X;\mathbb{Z}/2)$ for any $X$ as sitting in a short exact sequence

$$ 0 \to H^*(X;\mathbb{Z})/2 \to H^*(X;\mathbb{Z}/2) \to \text{2-tors in } H^{*+1}(X;\mathbb{Z})\to 0 $$

which is (noncanonically) split since all terms are $\mathbb{F}_2$-vector spaces. This allows you to describe cohomology with $\mathbb{Z}/2$-coefficients in terms of cohomology with $\mathbb{Z}$-coefficients.

EDIT: The below of course does not help with the ring structure and only describes the mod $2$ cohomology additively.

Since you wrote "2-torsion", I'm going to assume that $\mathbb{Z}_2$ refers to $\mathbb{Z}/2$ (rather than the $2$-adic integers, which I think is the preferred interpretation of the notation $\mathbb{Z}_2$ in those parts of algebraic topology close to algebra nowadays).

In that case, the short exact sequence $0\to \mathbb{Z} \to \mathbb{Z}\to \mathbb{Z}/2\to 0$ induces a long exact sequence on cohomology groups, exhibiting $H^*(X;\mathbb{Z}/2)$ for any $X$ as sitting in a short exact sequence

$$ 0 \to H^*(X;\mathbb{Z})/2 \to H^*(X;\mathbb{Z}/2) \to \text{2-tors in } H^{*+1}(X;\mathbb{Z})\to 0 $$

which is (noncanonically) split since all terms are $\mathbb{F}_2$-vector spaces. This allows you to describe cohomology with $\mathbb{Z}/2$-coefficients in terms of cohomology with $\mathbb{Z}$-coefficients.