On equivariant formal deformation theory

Using the set-up of deformation categories of Talpo and Vistoli, we re-interpret and generalize, in the context of cartesian morphisms in abstract categories, some results of Rim concerning obstructions against extensions of group actions in infinitesimal deformations. Furthermore, we observe that finite \'etale coverings can be infinitesimally extended and the resulting formal scheme is algebraizable. Finally, we show that pre-Tango structures survive under pullbacks with respect to finite, generically \'etale surjections $\pi:X\rightarrow Y$, and record some consequences regarding Kodaira vanishing in degree one.