Quotient and blow-up of automorphisms of graphs of groups

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 show that 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-groups isomorphisms $H_{i}: \cal{G}_{i} \rightarrow \cal{G}_{i}$ depends crucially on the $H_{i}$-conjugacy class of the correction term $\delta(\bar{E}_{i})$ for any edge $\bar{E}_{i}$ of $\bar{\cal{G}}$, where $H$-congjugacy is a new but natural concept introduced here. As an application we obtain a criterion as to whether a partial Dehn twist can be blown up relative to local Dehn twists to give an actual Dehn twist.