Minhyong's comments indicate the issue here. If I want to come up with an extension unramified outside $p$ then why not look at the 2-dimensional mod $p$ representation attached to the $\Delta$ function? This works for all but a very small set of $p$, where either the mod $p$ representation is degenerate, or $p$ is so small that $GL(2,p)$ is solvable anyway. For example the semisimple mod 2 representation attached to the $\Delta$ function is trivial. Gross' question was how to deal with this small set of primes.
For these small primes one can try other level 1 forms, of course. For example $p=691$ is a funny case where the representation attached to Delta is reducible, but for $p=691$ you can just use the level 1 weight 16 form instead. The problem with the smaller primes is harder to deal with, because e.g. a modular mod 2 representation unramified outside 2 must be reducible by an old theorem of Tate (look at bounds on discriminants -- the argument is delicate). So one has to try and look elsewhere. The trick with $p=2$ is due to Lassina Dembélé and you can get the paper at his website
Classical modular forms don't cut the mustard, so one seeks to try the same trick with Hilbert modular forms defined over a totally real field ramified only at 2. Such totally real fields are not hard to find, so the issue is now the following computational one -- how to compute the level 1 forms? Dembélé did this, and found an explicit example which gave a Galois representation into $GL(2,k)$ with $k$ of size 8 if I remember correctly (I have to get the kids out of bed -1 minutes ago so can't check any more details).