Probably the recent paper by Herpel and Stewart here helps to settle your basic question positively. Though the correspondence between subgroups of Lie groups and Lie algebras in the classical situation (or more generally for algebraic groups in characteristic 0) works quite well, it usually breaks down a lot in prime characteristic. But after going through all the work to classify semisimmple and then reductive algebraic groups and especially to understand their internal structure, it turns out that many aspects of the correspondence do work (most of the time) but usually with exceptions.
This isn't easy to predict, but for example the "smoothness" property which makes global and infinitesimal normalizers correspond is worked out for "fairly good" primes in their paper. Here the basic situation is the same as in your question: a pair consisting of a reductive group $G$ and a reductive subgroup $G'$. Denote the respective Lie algebras by $\mathfrak{g}$ and $\mathfrak{g'}$. As in characteristic 0, the Lie algebra normalizer (and centralizer) are the Lie algebras of the group normalizer (and centralizer), provided you avoid some primes for the root systems involved. Here it's easiest to start with a reductive group such as $\mathrm{GL}_n$ whose derived group is simple, to avoid complications with multiple root sytems.
In all characteristics, the automorphism group of a (say) simple algebraic group is easily understood' in particular, its identity component is just the adjoint group. Combining this with the Herpel-Stewart results in their $\S3$, I think you arrive at a positive answer to your question. This doesn't really involve representation theory, fortunately, where there are still many unknowns. In particular, it's apparently independent of thinking about the adjoint representation. But it does involve a lot of heavy lifting and has its own prerequisites.