Here is an alternate suggestion which should yield a reasonable upper bound on the size of $G_2$, and may provide the basis of an estimate for the size of $G_3$.
Any member of $G_2$ will be contained in the following class $C$ of graphs on 13 vertices: 5 vertices, which I call the core, will form one of the subgraphs listed in $G_1$, with 8, 10, or 12 edges going from the same 4 vertices of the core to the 8 vertices I call the rim, with an additional 0 up to 6 edges between the vertices of the rim, with every vertex having degree at most 4, and every vertex in the rim being adjacent to at least one of the 4 outer vertices of the core.
It is mildly tedious but not hard to list up to isomorphism those members of $C$ with 0 edges in the rim; there are less than 50 such representatives, which I shall group into a subclass called $C0$. Now let $i$ be an integer in the range from 0 to 5 and suppose we have the class $Ci$ of graphs which contain all isomorphism types (possibly with some duplication) of members of $C$ which have $i$ edges in the rim. Add a single edge in all possible ways to each member of $Ci$ and determine which such graphs are isomorphic, and this will form the class $Cj$, where $j=i+1$. Some candidates made this way will have to be rejected if, e.g., a vertex gets degree 4 or more.
I suspect that $C1$ will have fewer than 100 members, and that each subclass will have less than 10 times as many members as the previous subclass, except for $C5$ and $C6$, because of the symmetries involved. There will also be conditions to check on the small neighborhoods of each vertex, and there will be interesting restrictions when passing to $G_3$ which may exclude some members of $C$ from being in $G_2$.
I will attempt an exact enumeration of $C0$, $C1$, and $C2$ by hand. I make this post because I am not ready to write or borrow a graph isomorphism subroutine to implement in my current computing environment. This task of enumerating the $Ci$ by computer should be pie for anyone proficient in graph enumeration. Even having good estimates for $C3$ and $C4$ will be of use, should someone wish to take on a limited version of the task.
Gerhard "Ask Me About System Design" Paseman, 2012.01.11