We prove that for any group G, \(\pi _2^S(K(G,1))\), the second stable homotopy group of the Eilenberg–Maclane space K(G, 1), is completely determined by the second homology group \(H_2(G, \mathbb {Z})\). We also prove that the second stable homotopy group \(\pi _2^S(K(G,1))\) is isomorphic to \(H_2(G, \mathbb {Z})\) for a torsion group G with no elements of order 2 and show that for such groups, \(\pi _2^S(K(G,1))\) is a direct factor of \(\pi _{3}(SK(G,1))\), where S denotes suspension and \(\pi _2^S\) the second stable homotopy group. For radicable (divisible if G is abelian) groups G, we prove that \(\pi _2^S(K(G,1))\) is isomorphic to \(H_2(G, \mathbb {Z})\). We compute \(\pi _{3}(SK(G,1))\) and \(\pi _2^S(K(G,1))\) for symmetric, alternating, dihedral, general linear groups over finite fields and some infinite general linear groups G. For all finite groups G, we obtain a sharp bound for the cardinality of \(\pi _2^S(K(G,1))\).