Non-split extension of an extraspecial 3-group by SL(2,3)

Question

Let $Q$ be an extraspecial $3$-group of order $3^5$. I am curious about the existence of a group $G$ satisfies the following condition.

(1) $G$ is a non-split extension of $Q$ by the special linear group $\mathrm{SL}(2,3)$ where $Q$ is normal in $G$;

(2) If $P$ is a Sylow $3$-subgroup of $G$, then every element of order $3$ in $P$ is contained in $Q$.

(3) The action of Sylow $2$-subgroup of $G$ on $Q/Z(Q)$ is Frobenius.

Also, it will be great if someone could construct this group by GAP. Any explanation is appreciated.

Answer 1

Start with the second remark about groups of 1944:

gap> l:=AllSmallGroups(Size,1944,x->Size(DerivedSubgroup(x)),648);
[ <pc group of size 1944 with 8 generators>,
  <pc group of size 1944 with 8 generators>,
  <pc group of size 1944 with 8 generators>,
  <pc group of size 1944 with 8 generators> ]

Form a trivial module for 8 generators over $\mathbb{F}_3$, and construct extensions for the first group:

gap> m:=TrivialModule(8,GF(3));;
gap> e:=Extensions(l[1],m);;

Now we write a function to test for the conditions you require. For each extension we retroactively identify the group $Q$ (and make sure it is unique), we then test the 3 conditions

Testfun:=function(g)
local q,nat,a;
  # find q
  q:=List(GQuotients(g,SL(2,3)),Kernel);
  if Length(q)>1 then Error("more than one possible q");fi;
  q:=q[1];
  # Non-split
  if Length(ComplementClassesRepresentatives(g,q))>0 then return false;fi;
  # No outside element of order 3
  if ForAny(Difference(SylowSubgroup(g,3),q),x->Order(x)=3) then return false;fi;
  # Frobenius action of 2-Sylow
  nat:=NaturalHomomorphismByNormalSubgroup(g,Centre(q));
  a:=Action(Image(nat,SylowSubgroup(g,2)),Image(nat,q));
  if Size(a)<=8 or not IsTransitive(a) then return false;fi;
  return true;
end;

and then test the candidate extensions:

gap> e:=Filtered(e,Testfun);
[  ]

Doing do for the 4 candidate groups does not find a single example, so (unless I've made a mistake in interpreting your conditions) no such group exists.