$\newcommand{\p}{\mathcal{P}}$Let *G* be a group, and let $G_n$ denote the wreath product $G^n \rtimes \Sigma_n$. 

There seems to be a notion of a PROP $\p$ where the role of the symmetric groups $\Sigma_n$ is instead played by the groups $G_n$. An example would be $G=SO(N)$ and $\p$ the PROP associated to the framed little *N*-disc operad. Here $\p(1,n)$ has an action of $G_1^{op} \times G_n$ -- the copy of $G_1^{op}$ rotates the entire disc "counterclockwise" (i.e. an element $g \in SO(N)$ acts via $g^{-1}$), and the action of $G_n$ on $\p(1,n)$ is the evident one. The gluing maps are then suitably equivariant under this simultaneous action of $G$ on the input/output legs, so to speak.

Is there a standard name for this kind of PROP/operad? Cf. how one calls an operad where $\Sigma_n$ has been replaced by $B_n$ a braided operad. I would also be happy to hear of any paper where this kind of gadget has been defined and/or studied.