  In [F. Xu, On the cohomology rings of small categories, J. Pure Appl. Algebra
212 (2008), 2555-2569], Xu constructs a LHS-spectral sequence for target
regular extensions of small categories. We extend this construction to
ext-groups and construct a similar spectral sequence for source regular
extensions (with right module coefficients). As a special case of these
LHS-spectral sequences, we obtain three different versions of Slominska's
spectral sequence for the cohomology of regular EI-categories. We show that
many well-known spectral sequences related to the homology decompositions of
finite groups, centric linking systems, and the orbit category of fusion
systems can be obtained as the LHS-spectral sequence of an extension.
