If a group $G$ acts on a group $\Gamma$ (on the left, say) by group automorphism
then there is a semidirect product group whose underlying set is the product $\Gamma \times G$ but whose multiplication is twisted by $\rho$:
for $\delta, \gamma \in \Gamma,\; h,g \in G$, where $^h \gamma$ denotes the result of acting with $h$ on the left on $\gamma$.
This is called in group theory the semidirect product and written $\Gamma \rtimes \, G$.
There is a projection morphism $p:\Gamma \rtimes \, G \to G$ , $(\gamma, g) \to g$. A section $s$ of this can be identified with a derivation $d$, i.e. $d$ satisfies $d(h g) = (d h) \,^h (d g)$.
Let $H$ be any group. A decomposition of $H$ as an internal semidirect product consists of a subgroup $\Gamma$ and a normal subgroup $G$, such that every element of $H$ can be written uniquely in the form $\gamma g$, for $\gamma \in \Gamma$ and $g \in G$.
The internal and external concepts are equivalent. In particular, any (external) semidirect product $\Gamma \rtimes G$ is an internal semidirect product of the images of $\Gamma$ and $G$ in it.
The definitions above are not symmetric in left and right; since the first definition begins with a left action, we may call it a left semidirect product. Then a right semidirect product is given by an action on the right, or internally by the requirement that every element can be written in the form $g \gamma$.
However, right and left semidirect products are equivalent. Essentially, this is because any left action $(h,g) \mapsto {}^h{g}$ defines a right action $(g,h) \mapsto g^h \coloneqq {}^{h^{-1}}g$ and vice versa.
It is useful to generalise this to the case $\Gamma$ is a groupoid. This occurs if for example $\Gamma = \pi_1 X$ where $X$ is a (left) $G$-space.
So if $X=Ob(\Gamma)$, then $\Gamma \rtimes \, G$ has object set $X$ and a morphism $y \to x$ is a pair $(\gamma,g)$ such that $\gamma: y \to g x$ in $\Gamma$. The composition law is then given again by
if $(\delta, h): z \to y$, so that $\delta: z \to h y$ in $\Gamma$.
If $\Gamma$ is a discrete groupoid, and so identified with $X$, then we get $X \rtimes \, G$ which is the action groupoid of the action. In this case the projection $p: X \rtimes \, G \to G$ is a covering morphism of groupoids, i.e. any $g \in G$ has a unique lifting with given initial point. Note that if $Y \to X$ is a covering map of spaces, then the induced morphism of fundamental groupoids is a covering morphism of groupoids. If $q: H \to \pi_1 X$ is a covering morphism of groupoids, and $X$ admits a universal covering map, then there is a topology on $Y=Ob(H)$ such that $H \cong \pi_1 Y$. In this way, the category of covering maps of $X$ is equivalent to the category of covering morphisms of $\pi_1 X$.
The utility of the more general construction is that there is notion of orbit groupoid $\Gamma //G$ (identify any $\gamma$ and $^g \gamma$) and it is a theorem that the orbit groupoid is isomorphic to the quotient groupoid
where $N$ is the normal closure in $\Gamma \rtimes \, G$ of all elements $(1_x,g)$. Details are in the book reference below (but the conventions are not quite the same).
Semidirect product groups $A \rtimes_\rho G$ are precisely the split group extensions of $G$ by $A$. See at group extension – split extensions and semidirect product groups.
For $U(1) = \mathbb{R}/\mathbb{Z}$ the circle group, the automorphism group is
where the nontrivial element in $\mathbb{Z}_2$ acts on $\mathbb{R}$ by multiplication with $-1$. Write $\rho_{aut} : U(1) \times \mathbb{Z}_2 \to U(1)$ for the automorphism action. The corresponding semidirect product group is the group extension
where the group operation is given by
A general survey is in
Lecture notes include
Relevant textbooks include
R. Brown, Topology and groupoids, Booksurge 2006.
P. J. Higgins and J. Taylor, The Fundamental Groupoid and Homotopy Crossed Complex of an Orbit Space, in K.H. Kamps et al., ed., Category Theory: Proceedings Gummersbach 1981, Springer LNM 962 (1982) 115–122.