Call a finite group $A$ affine if it has a normal, self-centralizing, complemented, elementary abelian subgroup $V$. Such a group $A$ is a semi-direct product $G\ltimes V$ where $V$ is a vector space of dimension $n$ over $\mathbb{F}_p$ and $G$ is a group of matrices in $\operatorname{GL}(n,p)$. The elements of $A$ can be written as matrices $\left(\begin{smallmatrix}g& v \\ 0& 1 \end{smallmatrix}\right)$, where $g \in G$, $v \in V$ and $0,1$ are row vectors of the appropriate length. Conversely, given $G ≤ \operatorname{GL}(V)$, $V$ a vector space over $\mathbb{F}_p$, $A=G\ltimes V$ is an affine group.
An example is the full affine group, $\operatorname{AGL}(n,p)$ where $G$ is $\operatorname{GL}(n,p)$ and $V$ is $\mathbb{F}_p^n$, that is $A$ is all $(n+1)×(n+1)$ matrices of the form $\left(\begin{smallmatrix}g& v \\ 0& 1 \end{smallmatrix}\right)$ where $g$ is $n×n$ invertible, $v$ is anything, $0$ is a zero vector, and $1$ is just a $1×1$ identity matrix.
$V$ becomes a $G$-module and its $G$-module structure has a large influence on the group theoretic structure of $A$. In particular, $V$ contains no "trivial" (central) summand as a $G$-module iff $A$ is centerless.
Supposing $A$ is centerless, what conditions on $V$ ensure $A$ is a complete group, that is, so that $A$ is also "outerless"?
See the previous questions: