Maybe this is just a very fundamental problem, but I am not too sure the answer. It is well-known that $SL(2,5)$ is contained in $SL(2,q)$ iff $q$ is odd and $5\mid q(q^2-1)$. My question is whether we can always say $SL(2,5)$ is irreducible on the vector space $GF(q)^2$ ?

absolutelyirreducible, i.e. irreducible in its natural action on the the 2-d vector space over $k$, the algebraic closure of $GF(q)$. Again, the way to see this is using @Aakumadula's comment - the stabilizer in $GL_2(k)$ of a $1$-dimensional subspace is solvable and $SL(2,5)$ is not. – Nick Gill Jan 15 '13 at 10:04