The notion "holomorph" was introduced in Maria S. Voloshina's Ph.D. thesis On the Holomorph of a Discrete Group (available at https://arxiv.org/abs/math/0302120). It is defined as follows:
Let $G$ be a group and let $\mathrm{Aut}(G)$ be the automorphism group of $G$. The holomorph of $\mathrm{Hol}(G)$, is defined as follows:
- As a set, $\mathrm{Hol}(G)=\mathrm{Aut}(G)\times G$;
- For each $x,y\in G$ and $f,g\in\mathrm{Aut}(G)$, the multiplication on $\mathrm{Hol}(G)$ is defined by $$(f,x)\cdot(g,y)=(fg, g^{-1}(x)y)\text{.}$$
The author points out that there is a split exact sequence $$1\to G\to\mathrm{Hol}(G)\stackrel{\leftarrow}{\rightarrow}\mathrm{Aut}(G)\to1\text{.}$$
What I am confused about is, if there is such a split exact sequence, doesn't the splitting lemma imply that $\mathrm{Hol}(G)$ is isomorphic to $G\times\mathrm{Aut}(G)$?
This is an answer to the question that seems to have already been resolved in the comments.
As Prof. Conrad noted, the holomorph $\text{Hol}(G)$ of a group $G$ is isomorphic to the semidirect product $G\rtimes_\sigma \text{Aut}_{\text{Grp}}(G)$, where $\text{Aut}_{\text{Grp}}(G)$ is the group of group automorphisms of $G$, and $\sigma=\text{id}_{\text{Aut}_{\text{Grp}}(G)}:\text{Aut}_{\text{Grp}}(G)\to \text{Aut}_{\text{Grp}}(G)$ is the standard action.
An example of this group (interpreted at least measurably, so that the automorphisms are required to be at least Borel measurable; see e.g. Measurable group homomorphisms are continuous) when $G$ is commutative is the so-called affine group; one can similarly consider affine transformations of tori; see Example of a continuous affine group action, Question about Stillwell Naive Lie theory exercise 4.6.3, Dynamics on the torus. The second discussion is especially relevant. Indeed, note that $\text{Hol}(\mathbb{R})$ is isomorphic to
$$\mathbb{R}\rtimes \text{GL}(1,\mathbb{R})\cong (\mathbb{R},+)\rtimes (\mathbb{R}\setminus\{0\},\cdot), $$
so that both groups involved are abelian, and there is a short exact sequence
$$0\to\mathbb{R}\to \text{Hol}(\mathbb{R})\to \text{GL}(1,\mathbb{R})\to 1,$$
however this sequence is not in the category of abelian groups (it's in the category of possibly nonabelian groups).
Finally as a historical comment, the notion of the "holomorph of a group" goes at least to Burnside's 1897 book Theory of Groups of Finite Order (available at https://archive.org/details/cu31924086163726).