Restriction of automorphism to subgroup not implies automorphism

Statement

We can have a group $G$, a subgroup $H$, and an automorphism $\sigma$ of $G$ such that $\sigma (H)\subseteq H$, but $\sigma (H)$ is not equal to $H$. In other words, $\sigma$ restricts to an endomorphism of $H$ (which is necessarily an injective endomorphism), but the restriction is not an automorphism of $H$.

Related facts

• Restriction of automorphism to subgroup invariant under it and its inverse is automorphism
• Restriction of inner automorphism to subgroup not implies automorphism

Proof

Example of the integers and the rationals

Let $G$ be the group $(\mathbb {Q} ,+)$, i.e., the additive group of rational numbers. Let $H$ be the subgroup $(\mathbb {Z} ,+)$, i.e., the additive group of integers. Consider the automorphism $\sigma$ of $\mathbb {Q}$ given by $x\mapsto 2x$, i.e., it sends every rational number to its double.

$\sigma$ is clearly an automorphism of $\mathbb {Q}$, and $\sigma$ sends $\mathbb {Z}$ to within itself: the restriction of $\sigma$ to $\mathbb {Z}$ is the map that sends every integer to its double. The restriction of $\sigma$ to $\mathbb {Z}$ is not an automorphism of $\mathbb {Z}$: the image $\sigma (\mathbb {Z} )$ is the proper subgroup comprising even integers.