Given a quiver automorphism with nice properties, we give a presentation of the fixed point subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and Kleshchev between the cyclotomic Hecke algebra of type $G(r, 1, n)$ and the cyclotomic quiver Hecke algebra of type $A$, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type $G(r, p, n)$ which looks very similar to the one of a cyclotomic quiver Hecke algebra. In addition, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki–Koike algebras.