Repertory of the different sorts of operads Many different types of operads have emerged in recent years (symmetric, shuffle, cyclic, anticyclic, coloured, etc.).
I would like, for any of these, list the following data:


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*Description of the free object (combinatorial objects involved, expression for the composition maps, expression for auxiliary map (as e.g., the action of the symmetric group or the cyclic action);

*Given a collection $G$ of generators and its generating series $G(t)$, give an expression for the Hilbert series $H_G(t)$ of the operad generated by $G$;

*Give some examples of worthwile and well-studied operads that fit into this kind of operads.


References are naturally welcome but I would like to create here a big list (with one kind of operad by answer if possible).
 A: One operad that's popped up in my work is the kind of operad that has the same formal properties as the endomorphism operad of a space $X$, provided $X$ is equipped with an action of a topological group $G$.  Level $n$ of the endomorphism operad is just the space of maps $Map(X^n, X)$, but we think of this space as one with an action of the group
$$ G \times (\Sigma_n \ltimes G^n) \equiv G \times (\Sigma_n \wr G). $$
I sometimes denote this group $\Sigma^*_n \wr G$, and the family of groups $\Sigma^* \wr G$. I've been calling these operads $\Sigma^* \wr G$-operads as I thought the name was kind of harmless and more or less descriptive. 
The free objects over this operad look like a disjoint union of rooted trees, where the vertices of the trees are decorated by points in the generating $\Sigma^*_k \wr G$-spaces.  There is an equivalence relation on these decorated trees, generated by one relation for each edge of the trees -- corresponding to the endomorphism operad's equivariance. 
These types of operads come up (and are useful) with $G$ being various orthogonal groups, or diffeomorphism groups of balls (or other automorphism groups of balls) for certain embedding spaces, like spaces of knots.  For classical knots, like knots in the 3-sphere, there is an almost free operad of this type that describes the homotopy-type of the space of knots completely, up to the computation of certain (finite) symmetry groups of some hyperbolic links in $S^3$.  The operads I'm talking about I call splicing operads.  These are subspaces of $Map(X^n,X)$ but in this case, $X$ is a ball, and the maps are smooth maps with various restrictions on them (to get knots of various types). 
