The Hecke algebra of $G(r,p,n)$ can be seen as the fixed point subalgebra of the Hecke algebra of $G(r,1,n)$ (also known as the Ariki-Koike algebra) for a certain automorphism $\sigma$. Using an isomorphism of Brundan and Kleshchev with a KLR algebra, we find an analogue of $\sigma$ defined from the KLR presentation. Moreover, it turns out that we can give a KLR-like presentation of the fixed point subalgebra.