The Gröbner h-bases under composition

Polynomial composition is the operation of replacing variables in a polynomial with other polynomials. H.Hong has resolved following problem: when does polynomial composition commute with Gröbner basis computation? The main question of this paper is: when does polynomial composition commute with Gröbner h-bases computation? And we obtain some interesting results. These have some natural application in the computation of Gröbner bases of composed polynomials which often appear in real life problems.