For bounded surfaces the mapping class group is isomorphic to the boundary-fixing automorphisms of the fundamental groupoid.
Mapping class groups of covers with boundary and braid group embeddings.Algebraic & Geometric Topology, 20:239–278
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The Dehn-Nielsen-Baer Theorem for Bounded Surfaces
For bounded surfaces the mapping class group is isomorphic to the boundary-fixing automorphisms of the fundamental groupoid.