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Let $\tilde f\colon(S^2,\tilde A)\to(S^2,\tilde A)$ be a Thurston map and let$M(\tilde f)$ be its mapping class biset: isotopy classes rel $\tilde A$ ofmaps obtained by pre- and post-composing $\tilde f$ by the mapping class groupof $(S^2,\tilde A)$. Let $A\subseteq\tilde A$ be an $\tilde f$-invariantsubset, and let $f\colon(S^2,A)\to(S^2,A)$ be the induced map. We give ananalogue of the Birman short exact sequence: just as the mapping class group$\mathrm{Mod}(S^2,\tilde A)$ is an iterated extension of $\mathrm{Mod}(S^2,A)$by fundamental groups of punctured spheres, $M(\tilde f)$ is an iteratedextension of $M(f)$ by the dynamical biset of $f$.  Thurston equivalence of Thurston maps classically reduces to a conjugacyproblem in mapping class bisets. Our short exact sequence of mapping classbisets allows us to reduce in polynomial time the conjugacy problem in$M(\tilde f)$ to that in $M(f)$. In case $\tilde f$ is geometric (eitherexpanding or doubly covered by a hyperbolic torus endomorphism) we show thatthe dynamical biset $B(f)$ together with a "portrait of bisets" induced by$\tilde A$ is a complete conjugacy invariant of $\tilde f$.  Along the way, we give a complete description of bisets of $(2,2,2,2)$-mapsas a crossed product of bisets of torus endomorphisms by the cyclic group oforder $2$, and we show that non-cyclic orbisphere bisets have no automorphism.  We finally give explicit, efficient algorithms that solve the conjugacy andcentralizer problems for bisets of expanding or torus maps.

Algorithmic aspects of branched coverings III/V. Erasing maps, orbispaces, and the Birman exact sequence