  We present an explicit structure for the Baer invariant of a finitely
generated abelian group with respect to the variety
$[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$, for all $c_2\leq c_1\leq 2c_2$. As a
consequence we determine necessary and sufficient conditions for such groups to
be $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable. We also show that if
$c_1\neq 1\neq c_2$, then a finitely generated abelian group is
$[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable if and only if it is capable.
Finally we show that $\mathfrak{S}_2$-capability implies capability but there
is a finitely generated abelian group which is capable but is not ${\mathfrak
S}_2$-capable.
