  We introduce and analyze parallelizable algorithms to compress and accurately
reconstruct finite simplicial complexes that have non-trivial automorphisms.
The compressed data -- called a complex of groups -- amounts to a functor from
(the poset of simplices in) the orbit space to the 2-category of groups, whose
higher structure is prescribed by isomorphisms arising from conjugation. Using
this functor, we show how to algorithmically recover the original complex up to
equivariant simplicial isomorphism. Our algorithms are derived from
generalizations (by Bridson-Haefliger, Carbone-Rips and Corson, among others)
of the classical Bass-Serre theory for reconstructing group actions on trees.
