Does Higman's embedding theorem hold inside group varieties? Suppose $\mathfrak{U}$ is a variety of groups. Let's define $F_n(\mathfrak{U})$ as relatively free groups in $\mathfrak{U}$.
Suppose $G \in \mathfrak{U}$ 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.
This question on MSE
 A: Kharlampovich proved that there exist two finitely based varieties of solvable groups   $ {\mathfrak A} \subset {\mathfrak B}$ such that the word problem is not solvable in the groups f.p. in the smaller variety  but solvable in the groups that are f.p. in the bigger variety (the result can be found in our joint survey "Algorithmic problems in varieties", see also Kharlampovich, O. G.
The word problem for solvable Lie algebras and groups.
Mat. Sb. 180 (1989), no. 8, 1033–1066, 1150; translation in
Math. USSR-Sb. 67 (1990), no. 2, 489–525). Now if you take a group $G$ that is finitely presented in the smaller variety and has undecidable word problem then $G$ is finitely generated, recursively presented in the class of all groups since the varieties are finitely based, but cannot be embedded into any group $H$ which is finitely presented in the bigger variety. So variety ${\mathfrak B}$ does not have the property in the OP.   Varieties with that property are sometimes called Higman varieties. In my comment above I mentioned some known Higman varieties in addition to those in the OP. It is not known if the variety of solvable groups of class $\le 3$ is Higman. In fact all known Higman varieties have been mentioned in the OP and in my comment.
