Recognizing action of semidirect product I've been looking at some texts in representation theory and I see instances where the symmetric group $S_n$ and some other group, e.g., $GL(V_1) \times \ldots \times GL(V_n)$, act on a space. The author seems to just take for granted that there is an action of $(GL(V_1) \times \ldots \times GL(V_n)) \ltimes S_n$. My question is: is there always an action of the semidirect product? If not, is there an intuitive way to recognize when two actions allow an action of the semidirect product? 
In the case of a direct product, the answer is that the two commute.
 A: I'll assume all of your $V_i$ are in fact the same vector space $V$, or else I don't understand what your notation means. Then $S_n$ acts on $G \times \dots \times G$ by permuting the factors (in your example $G = GL(V)$ but this argument applies more generally), and your semidirect product is a wreath product
$$G \wr S_n = (G \times \dots \times G) \rtimes S_n.$$ 
You can think of elements of the wreath product as being "permutation matrices" where each nonzero entry in the permutation matrix, instead of being a $1$, is an element of $G$. If $G$ acts on an object $X$, then the wreath product naturally acts on the "$n^{th}$ power" of $X$
$$X^{\otimes n} = X \otimes \dots \otimes X$$ 
where $\otimes$ is any symmetric monoidal structure, such as the direct sum or tensor product of vector spaces, or the disjoint union or cartesian product of sets. The idea is that the copies of $G$ act "componentwise" on each copy of $X$ "individually" while the $S_n$ act by permuting the $X$. 
You can verify that this works by verifying that the defining relation of the semidirect product holds. For me the cleanest way to write it is as follows: if $N \rtimes H$ is a semidirect product, with $\varphi : H \to \text{Aut}(N)$ the defining action (which here is by permutation), then the defining relation is
$$hnh^{-1} = \varphi(h) n.$$
In general, if two groups $G$ and $H$ act on something, all you can conclude is that their free product $G \ast H$ also acts. To get a semidirect product you need, among other things, a candidate action of one of the groups on the other. In the above case that candidate action is by permuting factors but in general it needs to be provided as extra data. 
