Try to find an example for the strictly $<$ case? Normal subgroup

Question

Here is the definition for the normal subgroup:

A subgroup of $H$ that is invariant under all inner automorphisms is called normal; also, an invariant subgroup.

$$\forall \phi\in \text{Inn}(G), \phi(H)\le H$$

My question:

Here the definition use $\phi(H)\le H$ instead of $\phi(H)= H$, because $H$ might be infinite, right? (if it is finite, then $\le$ becomes $=$)

I know for the infinite case like $G=Z$, but since it is Abelian, it is still $\phi(H)= H$. How to find an example for the infinite case, when it is strictly $<$ ?

Answer 1

The $\le$ should imply $=$,

Proof: Because if $\phi\in \text{Inn}(G)$, then $\phi^{-1}\in \text{Inn}(G)$, so we have

$$\phi^{-1}(H)\le H \Rightarrow H\le \phi(H)$$

Combining with $\phi(H)\le H$, we have $\phi(H)=H$