Let $X_G$ be the number of normal subgroups of a group $G$. Are there examples of finitely generated groups $G$ where it is consistent to have $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups? Also are there examples where $\aleph_0<X_G/\mathord\sim<2^{\aleph_0}$ where $N \sim M \iff G/N \cong G/M$?
Finitely generated group with $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups?
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