Can some finite group $G$ embed by conjugacy into $\operatorname{Sym}(H)$, for some $H \lhd G$?

Question

Can some finite group $G$ embed by conjugacy into $\operatorname{Sym}(H)$, for some $H \lhd G$?

What I could work out is the following.

Let $G$ be a finite group and $H \lhd G$. Then, the map $\psi: g \mapsto (h \mapsto ghg^{-1})$ is a homomorphism from $G$ to $\operatorname{Sym}(H)$, with $\operatorname{ker}(\psi)=\{g \in G \mid ghg^{-1}=h, \forall h \in H\}$. Therefore, $\psi$ is an embedding of $G$ into $\operatorname{Sym}(H)$ if and only if:

$$g \in G \setminus \{e\} \Rightarrow \exists h \in H \mid gh \ne hg \tag 1$$

Having said that, I can't conclude whether $(1)$ is false for every such pair $(G,H)$, or instead it does hold for some.

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Edit (based on the accepted answer)

$$\operatorname{ker}(\psi)=\{g \in G \mid ghg^{-1}=h, \forall h \in H\}=C_G(H)$$

so that: $(H \lhd G) \wedge (C_G(H)=\{e\}) \Rightarrow G \hookrightarrow S_{|H|}$.

Answer 1

You’re asking for groups $G$ with subgroups $H$ that have trivial centralizers in $G$. If you don’t require $H$ to be a proper subgroup, you are asking for groups with trivial centers, such as $S_n$ for $n ≥ 3$.

Since you mean proper normal subgroups: Take $A_4 ⊆ S_4$.