Suppose $(O, G, \alpha)$ is a triple where $O$ is some mathematical object, $G$ is a group and $\alpha : G \rightarrow Aut(O)$. Many different areas of mathematics study such triples. However, I only know of a couple of examples of a new object that encodes this system.

1. If $G$ is a group, $N$ a normal subgroup of $G$, $H$ a subgroup of $G$ and $\alpha(h)(n) = hnh^{-1}$, then one can encode $(N, H, \alpha)$ into the (outer) semidirect product $N \rtimes_\alpha H$.

2. If $O = \mathcal A$, a C$^*$-algebra, $G$ is a locally compact group and $\alpha : G \rightarrow Aut(\mathcal A)$ acts by $*$-automorphisms then $(\mathcal A, G, \alpha)$ is encoded as the crossed product algebra $\mathcal A \rtimes_\alpha G$. In very simple terms, this is accomplished by considering unitary representations of $G$. There is also a related construction in Statistical Mechanics called the covariance algebra.

My question then: are there any other semidirect/crossed product type constructions for such triples $(O, G, \alpha)$?