The nonabelian product modulo sum

It is shown that if $\{H_n\}_{n \in \omega}$ is a sequence of groups without involutions, with $1 < |H_n| \leq 2^{\aleph_0}$, then the topologist's product modulo the finite words is (up to isomorphism) independent of the choice of sequence. This contrasts with the abelian setting: if $\{A_n\}_{n \in \omega}$ is a sequence of countably infinite torsion-free abelian groups, then the isomorphism class of the product modulo sum $\prod_{n \in \omega} A_n/\bigoplus_{n \in \omega} A_n$ is dependent on the sequence.