I am relatively new to category theory, so only have a rough understanding of the technicalities behind operads. My understanding is that symmetric operads are defined so that they are "nicely" acted on by the symmetric group.
My question is: Is it possible to alter this action i.e. quotient the action so that a proper subgroup of the symmetric group acts on the operad?
I do not see how this would raise any issues with the definition of the symmetric operad (for example I think you still have equivariance), but am wondering if any technicalities would prevent me from doing this.
Yes. (This paper formulates extra structure on a sequence $G_0,G_1,\dots$ of groups to get a theory analogous to the theory of operads.)