**Edit:** I revise the  question based on the comment  conversations


Let $\mathcal{F}$ be  the set of  all equivalent classes  of  finite  groups under the "Isomorphism" equivalent relation.
We  define a  pseudo metric  $d$  on $\mathcal{F}$ as  follows:

$$d(G,H)= \inf \{Hd(\tilde {G}_{n},\tilde{H}_{n})\} $$

where  $\inf$  takes  over all arbitrary   isomorphic  copy $\tilde{G}_{n}$  and  $\tilde{H}_{n}$ of  $G$ and  $H$ in   $Gl(n,\mathbb{R})$, respectively. 
Furthermore $Hd$ is the   Hausdorff  distance   in $GL(n,\mathbb{R})$ based on  its standard left  invariant  metric.

The  definition of  this  metric  is  motivated  by the  Hausdorff Gromov  metric  on the  space  of  compact  Riemannian  manifolds.

>Is $d$   a  metric  on $\mathcal{F}$? If  the  answer  is  yes, we  denote by  $\bar{\mathcal{F}}$ the   completion of $\mathcal{F}$. What  can be  said about  an object $Z$ in  $\bar{\mathcal{F}}$? 

Can one  consider  the unit circle, in some reasonable  sense, as  an  objects in this completion?
Is there  a  natural group  structure  on every element $Z\in \bar{\mathcal{F}}$? Is there  a  natural topology on $Z$?

Is  $\bar{\mathcal{F}}$ a  compact space?