Let $H=G\rtimes \operatorname{Aut}G$ be the holomorph of $G$. Obviously $G$ embeds as $g\mapsto (g,1)$, and this embedding is normal. Composing this with any automorphism of $G$ is also a normal embedding; for the purpose of this question, I am not interested in distinguishing embeddings that only differ by an automorphism of $G$.
$G$ also embeds as $g\mapsto (g^{-1},\rho_g)$, where $\rho_g$ is conjugation by $g$, and this embedding is also normal. If and only if $G$ is abelian, $\rho_g=1,\forall g$, and the two embeddings become the same up to the automorphism $g\mapsto g^{-1}$.
What I am generally interested in is - are there other embeddings? Other normal embeddings? Are there any other "canonical" normal embeddings?
But this is a little soft, so let me ask a much more circumscribed question:
Let $H = Q_8\rtimes \operatorname{Aut}Q_8$, where $Q_8$ is the quaternion group. What are all the embeddings of $Q_8$ in $H$? Which are normal?