Homogeneous Gröbner bases under composition

Let K[x1,…,xn] be the polynomial ring over a field K in variables x1,…,xn. Let Θ=(θ1,…,θn) be a list of n homogeneous polynomials of same degree in K[x1,…,xn]. Polynomial composition by Θ is the operation of replacing xi of a polynomial by θi. The main question of this paper is: When does homogeneous polynomial composition commute with homogeneous Gröbner bases computation under the same term ordering? We give a complete answer: for every homogeneous Gröbner basis G, G○Θ is a homogeneous Gröbner basis under the same term ordering if and only if the composition by Θ is homogeneously compatible with the term ordering and Θ is a ‘permuted powering.’