A question regarding Steam Groups and members.

I don't think so

I actually remember a group who did some group competition where the admin was going to give a gift to someone who invited the most people. He had all this statistic about it like who invited who, who left the group out of those invited, and the number of people he/she invited in total.

The only way is from Group History that is right on Admin Tools -> "View Group History".
There you can see who left the group / who joined the group / who invited members in.

Thanks to everyone. The answer really is in View Group History. Thanks again :)