Restriction of automorphism to subgroup not implies automorphism

From Groupprops

Statement

We can have a group $G$ , a subgroup $H$ , and an automorphism $σ$ of $G$ such that $\sigma(H) \subseteq H$ , but $σ(H)$ is not equal to $H$ . In other words, $σ$ 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 $σ$ of $\mathbb{Q}$ given by $x \mapsto 2x$ , i.e., it sends every rational number to its double.

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