Consider a countable collection of particles located on a countable group, performing a critical branching random walk where the branching rate of a particle is given by a random medium fluctuating both in space and time. Here we study the case where the time-space random medium (called catalyst) is also a critical branching random walk evolving autonomously while the local branching rate of the reactant process is proportional to the number of catalytic particles present at a site. The catalyst process and the reactant process typically have different underlying motions.