Is there a big solvable subgroup in every finite group? Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H \cup \{x\} \rangle = L$.
Question: Is there a big solvable subgroup in every finite group?
Motivation: By Theorem A in M.Aschbacher and R.Guralnik, "Solvable generation of groups and Sylow subgroups of the lower central series", in every finite group $G$ there exists a pair of conjugate solvable subgroup $H_1,H_2 \leq G$ such that $\langle H_1 \cup H_2 \rangle = G$. So if $x \in G$ is such that $xH_1x^{-1} = H_2$ then $\langle H_1 \cup \{x\} \rangle = G$. In my question I am asking for a generalization of this conclusion.
 A: As a partial answer generalizing Pablo's example, the claim is true for a group with a $(B,N)$-pair for which $B$ is solvable. In this case, take $H:=B$. This recovers Pablo's example as $A_5$ acts doubly transitively on a set with 5 points, hence has a $(B,N)$-pair, with a solvable point stabilizer $B=A_4$. Other examples include e.g. general linear groups $GL_n(K)$.
See Wikipedia for the definition of a $(B,N)$-pair and the notation used in the sequel. 
Indeed, any subgroup $L$ satisfying $B\leq L \leq G$ is a standard parabolic subgroup, $L=P_X$
for some subset $X$ of the index set of the Dynkin diagram of $G$. Write $X$ as the union of its connected components, $X=\cup C_i$, and for each $i$ choose some $g_i$ in the big cell ${B_X}^+{B_X}^-$. Then $g:=g_1\cdots g_n$ is as required, i.e. $B_g:=\langle B,g\rangle = L$. To see this, note that $B_g$ again is a standard parabolic subgroup and by the choice of $g_i$, which isn't contained in a proper parabolic subgroup of $P_{C_i}$, $B_g \cap P_{C_i} = P_{C_i}$.
A: Michio Suzuki proved that every finite group is generated by a pair of conjugate solvable subgroups. See http://projecteuclid.org/download/pdf_1/euclid.hokmj/1381517825 (Open access). The subgroup $S$ which Suzuki exhibits is not a priori "big" in your sense (as far as I can see), but may suggest an approach to the question.
Later edit: Ah, I see that the existence part of Suzuki's result is no stronger than that of Aschbacher and Guralnick, it's just that his subgroup $S$ has more properties which may be useful from an inductive point of view.
