Linear occurrences of finite simple groups Let $S$ be a finite simple group. All representations below are over the complex numbers. 
Let 


*

*$d_0(S)$ be the smallest dimension of a faithful representation of $S$,

*$d_1(S)$ be the smallest dimension of a faithful representation of some central extension of $S$, and

*$d_2(S)$ be the smallest dimension of a faithful representation of some finite group admitting $S$ as quotient (or, equivalently, of some finite group admitting $S$ as Jordan-Hölder factor).


In a sense, $d_2(S)$ can be thought of the smallest dimension in which $S$ occurs linearly. On the other hand, $d_0$ and $d_1$ are well-documented, but, as far as I know, not $d_2$.
Trivially $d_0\ge d_1\ge d_2$, and the left-hand inequality can be strict: $d_0(\mathrm{Alt}_5)=3>2=d_1(\mathrm{Alt}_5)$. 

Does there exist $S$ with $d_1(S)>d_2(S)$?

I'd be interested as well by any information about $d_2$, including values for small non-abelian simple groups, or some particular families.
Note: in a naive attempt to show $d_1=d_2$, we could wonder if any finite group having $S$ as quotient admits a subgroup isomorphic to some central extension of $S$; a counterexample is pointed out by Derek Holt here. 
 A: Searching and starting from Noam's answer, I found some extra information:
the phenomenon pointed out by Noam on automorphism groups of extraspecial 2-groups seems to appear in: R.L. Griess, Jr. Automorphisms of extra special groups and nonvanishing degree 2 cohomology. Pacific J. Math 48 (73) 403-422. (Link, MR link)
The discussion about $d_1$ and $d_2$ seems to appear in:
W. Feit and J. Tits, Projective representations of minimum degree of group extensions, Canad. J. Math. 30 (1978) 1092-1102, and continues in P. Kleidman and M. Liebeck. On a theorem of Feit and Tits, Proc. AMS 107(2), 1989, 315-322. (Link, MR link)
Feit-Tits establish that if $d_2(S)<d_1(S)$, then $S$ is of Lie type over a finite field of characteristic 2, and in this case $d_2$ is a power of 2. Kleidman and Liebeck establish the precise list of those such groups of Lie type that indeed satisfy $d_2<d_1$, including the computation of $d_2$. 
In particular, it follows from this classification that the smallest dimension in which some finite simple groups ``occur" but not through a projective representation is 16, namely for the groups $\Omega_8^-(2)$, $\mathrm{Sp}_8(2)$, $\mathrm{Sp}_4(4)$.  
Besides, if we consider representations over an algebraically closed field of characteristic $p\ge 5$, defining $d_i^{(p)}$ in the same fashion: the picture is exactly the same. In char. 3 it's essentially the same up to a minor difference in the classification among those $S$ of Lie type over a field of characteristic 2. On the other hand in characteristic 2 it's dramatically different: $d_1^{(2)}(S)=d_2^{(2)}(S)$ for all $S$.
A: Seems that $S = O_{2n}^\pm(2)$ are examples of this for $n=5$, and probably for all $n \geq 5$.  Such $S$ is a Jordan-Hölder factor of the automorphism group of the extraspecial group $2^{1+2n}_\pm$, so $d_2 \leq 2^n$.  But the Schur multiplier is trivial, so $d_0 = d_1$, and the ATLAS of Conway et al. reports minimal faithful representations of dimensions $154$ for $O_{10}^-(2)$ and $155$ for $O_{10}^+(2)$, both larger than $2^5 = 32$.
