Look up Gelfand pairs. You want (G,H) to be a Gelfand pair. A sufficient condition is that each orbit on G/HxG/H be symmetric (viewed as a relation). This is called a symmetric Gelfand pair. Other sufficient conditions are known.
Additional information. My favorite book on Gelfand pairs is Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains by Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli.
Here they study Markov chains on $G/H$ using that the endomorphism algebra is commutative. Specific examples coming from symmetric groups and wreath products can be found in the book. Also Weilandt's classical book on permutation groups has a chapter on centralizer algebras and the connection with orbitals is made clear.