The action of the automorphisms of a formal group on its deformation space is crucial to understanding periodic families in the homotopy groups of spheres and the unsolved Hecke orbit conjecture for unitary Shimura varieties. We can explicitly pin down this squirming action by enriching it -- we may view the action as coming from an action on a moduli stack with rich geometric structure. We will explain necessary conditions for a moduli stack to model this action, construct moduli stacks which satisfy these conditions, and use our understanding of the group acting on these stacks to describe the action in general for all maximal finite subgroups for odd primes.