I think Abantecart has the session info of the admin so if you click on that storefront icon, it works from knowing that the new session to the store is also the admin (the devs can confirm how it's coded). Your browser needs to not be in anonymous/private mode.
If it's working without the https/ssl, then maybe it's the same thing that I ran into.
Specifically for me, in Firefox, the steps I would use was:
1) I would login to the admin side.
2) Click the storefront icon which opened a new tab to the store with the page which would partially work (no maintenance page, but not looking right).
3) Go back to my admin, go to System -> Settings -> Detail
4) Click SSL off, save
5) Click SSL on, save
6) In the new tab with my just opened storefront, refresh...
After that, it would work for me.