Subgroup Containing all Squares of Group Elements is Normal/Corollary

Corollary to Subgroup Containing all Squares of Group Elements is Normal
Let $G$ be a group.

Let $H$ be a subgroup of $G$ with the property that:
 * $\forall x \in G: x^2 \in H$

The quotient group $G / H$ is abelian.

Proof
From Subgroup Containing all Squares of Group Elements is Normal, $H$ is normal in $G$.

For every $x \in G$, we have:

Thus by definition $G / H$ is boolean.

The result follows by Boolean Group is Abelian.