Hey Radomir,
You've got it all correct but you're missing one thing. Your shell doesn't actually have access to the new group permissions yet. You have to log back in or instantiate a new shell so that you'll have access to the new group permissions.
You can also run the command newgrp after you change groups or group permissions, but realize all it's doing is instantiating a new shell! Keep in mind that this only is relevant to adding users to a group.
To prove this leave your old shell open that you used to add the group. You'll see you have no access to it.
Now open a new shell instance and check if you have access.
You should see things work as they should on the new shell.
Now go back to the old shell where things don't work and run newgrp.
All should work as you expected.
Don't go chmodding everything to 777 either!
Bryan