Is any variety generated by a recursively presented group always recursive? Let’s denote the minimal variety of groups (a class of all groups, that satisfy a given set of identities of equivalently a class of groups that is closed under subgroups, quotients and direct products) that contains a group $G$ as $Var(G)$, the relatively free group  of rank $n$ for the variety $\mathfrak{U}$ as $F_n(\mathfrak{U})$.

Is it always true, that if $G$ is recursively presented, then so is $F_n(Var(G))$?

Alternatively, this question can be formulated the following ways:

Is it always true that if $G$ is recursively presented, then $Var(G)$ can be defined by a recursively enumerable set of identities?


Is it always true that if $G$ can be embedded into a finitely presented group, then $F_n(Var(G))$ also can?

The third variant of the question is equivalent to the first two due to Higman embedding theorem
However, neither of these formulations gives me any idea how to approach this problem…

Yu. Kleiman constructed a finitely generated recursively presented group $G$ such that the variety $var(G)$ is not defined by a recursively enumerable set of identities. See the math review MR0688918.
