Holomorph ($\mathrm{Hol}(G)=G⋊\mathrm{Aut}(G)$)

The holomorph acts on the underlying set of a group by $x\mapsto g\alpha (x)$. Its multiplication is $(g,\alpha )(h,\beta )=(g\alpha (h),\alpha \beta )$. This is exactly the normalizer of the left regular permutation representation: a normalizing permutation fixing the identity is an group automorphism, and every normalizing permutation is a left translation followed by one. The embedded copy of $G$ is normal and the identity stabilizer is its group automorphism group.