This is about question $4.1.7$ from Dummit and Foote, and also related to my previous question.
The question is (summarised a bit):
Let $G$ be a transitive permutation group on a finite set $A$. A block $B\subset A$ is a set s.t. $\sigma(B)\cap B$ is either $\emptyset$ or $B$ for all $\sigma\in G$.
a) $B$ a block $\Rightarrow$ $G_a\leq G_B\leq G$ for all $a\in B$.
b) $B$ a block, $\sigma_1(B),\dots,\sigma_n(B)$ the distinct images of $B$ $\Rightarrow$ $\sigma_1(B),\dots,\sigma_n(B)$ are a partition of $A$.
c) A transitive group on $A$ is primitive if the only blocks are $A$ and $\{a\}$ for $a\in A$ (and some trivial exercise)
d) A transitive group on $A$ is primitive $\Leftrightarrow$ for all $a$: $G_a$ is a maximal subgroup of $G$
I've done these exercises, but I've never used the fact the $G$ has a trivial kernel, which is the only way in which I can imagine I would use the fact that $G$ is a permutation group.
If needed I can write out all the proofs, but I hope someone can immediately tell me which, if any, of these exercises depend on $G$ being a permutation group, and then I can probably find my mistake from there.
Statement (c) does not make sense (because $G$ and $G_a$ are not subsets of $A$ and so cannot be blocks). The other statements, however, are true and would remain true even if "Let $G$ be a transitive permutation group" were replaced by "Let $G$ be a group acting transitively". So the fact that you didn't need to assume $G$ has a trivial kernel does not signal a mistake on your part.