status: appeared in Adv.Math., here is MathSciNet bibtex:

@article {MR2200855,
 AUTHOR = {Grodal, Jesper and Smith, Stephen D.},
 TITLE = {Propagating sharp group homology decompositions},
 JOURNAL = {Adv. Math.},
 FJOURNAL = {Advances in Mathematics},
 VOLUME = {200},
 YEAR = {2006},
 NUMBER = {2},
 PAGES = {525--538},
 ISSN = {0001-8708},
 CODEN = {ADMTA4},
 MRCLASS = {20J06 (55P91 55R35)},
 MRNUMBER = {MR2200855},
}


\subjclass{Primary: 20J06; Secondary: 20J05, 55R35, 55P91}

\author[J.~Grodal]{Jesper Grodal}
\address{Department of Mathematics, University of Chicago, Chicago, IL 60637, USA}
\email{jg@math.uchicago.edu}

\author[S.~D.~Smith]{Stephen D.~Smith}
\address{Department of Mathematics (m/c 249),
         University of Illinois at Chicago,
	 851 S. Morgan, 
         Chicago, IL 60607-7045, USA}
\email{smiths@math.uic.edu}

\begin{abstract}
A collection $\cC$ of subgroups of a finite group $G$
can give rise to three different standard
formulas for the cohomology of $G$ in terms of either
the subgroups in $\cC$ or their centralizers or their normalizers.
We give a short but systematic study
of the relationship among such formulas
for nine standard collections $\cC$ of $p$-subgroups,
obtaining some new formulas in the process.
To do this, we exhibit some sufficient conditions on the poset $\cC$
which imply comparison results.
\end{abstract}