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In this paper we study the quotient and "blow-up" of graph-of-groups$\cal{G}$ and of their automorphisms $H: \cal{G} \rightarrow \cal{G}$. We showthat the existence of such a "blow-up" of $\bar{H}: \bar{\cal{G}} \rightarrow\bar{\cal{G}}$ relative to a given family of "local" graph-of-groupsisomorphisms $H_{i}: \cal{G}_{i} \rightarrow \cal{G}_{i}$ depends crucially onthe $H_{i}$-conjugacy class of the correction term $\delta(\bar{E}_{i})$ forany edge $\bar{E}_{i}$ of $\bar{\cal{G}}$, where $H$-congjugacy is a new butnatural concept introduced here. As an application we obtain a criterion as towhether a partial Dehn twist can be blown up relative to local Dehn twists togive an actual Dehn twist.

Quotient and blow-up of automorphisms of graphs of groups