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 1neq 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.

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