  In this paper we continue our study of property (RD) for Hecke pairs
initiated in (V. Shirbisheh, Property (RD) for Hecke pairs. Mathematical
Physics, Analysis and Geometry, vol. 15, no. 2, (2012), 173--192). We study the
behavior of property (RD) under different commensurability relations of
subgroups in Hecke pairs. As an application, we prove that if H is a normal
subgroup of a group G and K is a subgroup of G commensurable to H, then the
Hecke pair (G,K) has (RD) if and only if the quotient group G/H has (RD). This
is used to investigate an infinite number of non-elementary examples of Hecke
pairs with property (RD). In particular, we introduce a class of groups whose
all subgroups are almost normal and all such Hecke pairs have property (RD). We
also discuss property (RD) of certain Hecke pairs arising from group
extensions.
