A finite group acting on a complex vector space of dimension $n$ can be seen as acting on a real vector space of dimension $2n$ just by forgetting the complex structure of the space. My question is, if I am handed a real vector space $V$ of dimension $2n$, and a group $G$ acting on it, is there a test I can perform to determine if the action arose from a complex action in this way?
Sometimes it is easy to rule out: for example, if $G$ contains anything orientation-reversing, then clearly it doesn't arise in this way. Or if one knows enough about $G$ (abstractly as a group) to know it doesn't have any faithful $n$-dimensional representation. But I would like an if-and-only-if criterion:
Is there a test I can perform on the pair $G,V$ to determine whether the action of $G$ can be obtained by beginning with a complex $n$-dimensional representation and forgetting the complex structure of the vector space?
To make a little more precise what I mean by "can be obtained": if there is an element $J\in GL(V)$ that commutes with the action of $G$ and satisfies $J^2 = - I$, then the action of $G$ "can be obtained from a complex $n$-dimensional action by forgetting the complex structure", since one can regard $V$ as a complex vector space via the action of $J$. So the question is, if I am handed $G$ and $V$, is there a test for the existence of such a $J$?