  We set up a Grothendieck spectral sequence which generalizes the
Lyndon--Hochschild--Serre spectral sequence for a group extension $K\mono G\epi
Q$ by allowing the normal subgroup $K$ to be replaced by a subgroup, or family
of subgroups which satisfy a weaker condition than normality. This is applied
to establish a decomposition theorem for certain groups as fundamental groups
of graphs of Poincar\'e duality groups. We further illustrate the method by
proving a cohomological vanishing theorem which applies for example to
Thompson's group $F$.
