Equivariant non symmetric operads The definition of a symmetric $G$-Operad is basically a $G$ object in the category of symmetric operads. As far as I understand there is not a good notion of the non symmetric case. I would like to know what are the main drawbacks of defining it as a $G$ object in the category of non symmetric operads.
 A: I see no issues at all defining a $G$-equivariant non-symmetric operad $O$ as a $G$-object in the category of non-symmetric operads. Depending on your setting -- pointed or unpointed, reduced or non-reduced -- you might need to specify that the action fixes the unit in $O(1)$. For example, see Section 2.4 of my paper with Javier Gutierrez on $G$-equivariant symmetric operads. 
With this definition, $G$-equivariant non-symmetric operads would sit inside $G$-equivariant symmetric operads, just like the non-equivariant setting. There's lots of operadic questions you could investigate. I do agree that the homotopy theory would be less interesting, for the reason Denis pointed out (in the symmetric setting, you need to look at subgroups of $G \times \Sigma_n$, so without the $\Sigma_n$ actions you are left with only subgroups of $G$). Basically, it should work like it does for $G$-spaces. Anyway, don't let any of this stop you from investigating. I'm sure there's lots you can prove about $G$-equivariant non-symmetric operads!
