Let $\Omega$ be a signature (in the sense of universal algebra) and let $T(X)$ be the absolutely free $\Omega$-algebra with generators $X$. Let $G = Aut(T(X))$ be the group of automorphisms of $T(X)$. Let $\sim$ be the equivalence relation induced by $G$ (viewed as acting on $T(X)$). The relation $\sim$ is not in general a congruence. So the set of orbits $T(X)/G$ does not naturally carry the structure of a $\Omega$-algebra. What kind of structure does it naturally have?
To be a bit more concrete consider a propositional language. The signature is $\Omega = (\neg, \wedge)$ with $\mathsf{ari}(\neg) = 1$ and $\mathsf{ari}(\wedge) = 2$. $X$ is some countable set of propositional variables $p_1, p_2, \dots$. Hence $T(X)$ can be thought of as a set of propositional formulas and $Aut(T(X))$ is isomorphic to $Sym(X)$ (the group of permutations of $X$). When $p\neq q$ we have $p\sim q$ and $p\sim p$ but not $p\wedge p\sim p\wedge q$ (for $p, q\in X$). So $\sim$ is not a congruence. When we look at the orbit space, the orbits are things like: the set of propositional variables, conjunctions both of whose conjuncts are propositional variables such that the left conjunct is distinct from the right conjunct, conjunctions both of whose conjuncts are propositional variables whose left conjunct is identical to the right conjunct... and so on. Is there any nice or natural description of the structure of this orbit space?