This work examines the possible projectivity of 2-modular block parts of nonprojective Lefschetz characters over 2-local geometries of several sporadic groups. Previously known results on M12, J2, and HS are mentioned for completeness. The main new results are on the sporadic groups Suz, Co3, Ru, O'N, and He. For each group, the Lefschetz character is calculated, and its 2-modular block parts are examined for projectivity. In each case it is confirmed that a nonprincipal block part contains a nonprojective summand. The case of O'N is additionally found to have a nonprojective summand in its principal block part. Nineteen of the sporadic groups (including many previously known cases) are categorized into three classes based on projectivity properties of their Lefschetz characters. [ABSTRACT FROM AUTHOR]