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

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.