Let *G* be a simple, simply connected algebraic group defined over an algebraically closed field *k* of positive characteristic *p*. Let *σ* :*G* → *G* be a strict endomorphism (i.e., the subgroup *G*(*σ*) of *σ*-fixed points is finite). Also, let *G*_{σ} be the scheme-theoretic kernel of *σ*, an infinitesimal subgroup of *G*. This paper shows that the dimension of the degree *m* cohomology group H*m*(*G*(*σ*),*L*) for any irreducible *k**G*(*σ*)-module *L* is bounded by a constant depending on the root system Φ of *G* and the integer *m*. These bounds are actually established for the degree *m* extension groups
$ Ext^{m}_{G(\sigma )}(L,L^{\prime })$
between irreducible *k**G*(*σ*)-modules
$L,L^{\prime }$
, with a similar result holding for *G*_{σ}. In these Ext*m* results, the bounds also depend on the highest weight associated to *L*, but are, nevertheless, independent of the characteristic *p*.

We also show that one can find bounds independent of the prime for the Cartan invariants of *G*(*σ*) and *G*_{σ}, and even for the lengths of the underlying PIMs.