  This work examines the possible projectivity of 2-modular block parts of
non-projective Lefschetz characters over 2-local geometries of several sporadic
groups. Previously known results on $M_{12}$, $J_2$, and $HS$ are mentioned for
completeness. The main new results are on the sporadic groups $Suz$, $Co_3$,
$Ru$, $O'N$, and $He$. For each group, the Lefschetz character is calculated,
and the 2-modular blocks are examined for projectivity. In each case it is
confirmed that a non-principal block contains a non-projective summand. The
case of $O'N$ is additionally found to have a non-projective summand in its
principal block part.
