TITLE: Characteristic subgroups $\phi(H) \subseteq H$
QUESTION [7 upvotes]: My book uses this definition: 

Let $G$ be a group. A subgroup $H$ of $G$ is called a characteristic subgroup if $\phi(H) \subseteq H$ for all $\phi \in \operatorname{Aut}(G)$.

But after some googling around, it seems that the definition for a characteristic subgroup involves equality $\phi(H) = H$. 
Does  $\phi(H) \subseteq H$ $\Rightarrow$ $\phi(H) = H$ ???
I tried to multiply by $\phi ^{-1}$ to get $H \subseteq \phi^{-1}(H)$ but I'm not sure if I am allowed to that and I'm even more unsure if $\phi(H) \subseteq H$ and $H\subseteq \phi^{-1}(H)$ $\Rightarrow$ $\phi(H) = H$.
Any mathematical wisdom? Thank you.

REPLY [6 votes]: The definitions are equivalent.
The answer of blue answers the question you ask, but does not address the underlying question of whether the two definitions are equivalent. So I will do that here.
So, no, $\phi(H) \subseteq H$ does not imply $\phi(H) = H$. However:

If $\phi(H) \subseteq H$ for all $\phi\in\operatorname{Aut}(G)$ then $\phi(H)=H$ for all $\phi\in\operatorname{Aut}(G)$. (Note: The key point is the "for all".)

Clearly, this implies that the definitions are equivalent. To see that the above holds, note that the following both hold for all $\phi\in\operatorname{Aut}(G)$.

$\phi(H)\leq H$
$\phi^{-1}(H)\leq H$

However, the second point can be manipulated to give you that $H\leq\phi(H)$. Combining this with the first point implies that $\phi(H)=H$ for all $\phi\in\operatorname{Aut}(G)$, as required.