  We provide a direct and elementary proof of the equivalence between the weak
asymptotic homomorphism property for the pair of group von Neumann algebras
$L(H)\subset L(G)$ and the embedding into $H$ of the one sided quasi-normalizer
of the pair $H<G$, as it is stated by J. Fang, M. Gao and R. R. Smith in a
recent article, and we extend the result to triples of groups $H<K<G$.
