Suppose $\mathfrak{U}$ is a variety of groups. Lets define $F_n(\mathfrak{U})$ as relatively free groups in $\mathfrak{U}$
Suppose $G$ is a finitely generated group. We call $G$ finitely presented in $\mathfrak{U}$ iff $\exists n \in \mathbb{N}$ and finite $A \subset F_n(\mathfrak{U})$ such that $G \cong \frac{F_n(\mathfrak{U})}{\langle \langle A \rangle \rangle}$. We call $G$ recursively presented in $\mathfrak{U}$ iff $\exists n \in \mathbb{N}$ and recursively enumerable $A \subset F_n(\mathfrak{U})$ such that $G \cong \frac{F_n(\mathfrak{U})}{\langle \langle A \rangle \rangle}$.
My question is:
Is it true, that a finitely generated group is recursively presented in $\mathfrak{U}$ iff it is isomorphic to a finitely generated subgroup of a group finitely presented in $\mathfrak{U}$?
This fact is true for the varieties of abelian groups due to linear algebra, proved for the variety of all groups by Higman and for the Burnside varieties by Olshanski.
However, I do not know, whether it is true in general.