When you create as admin a ticket for a customer, assign the ticket to a group, the customer recieve an email with a url that is pointing to the ticket.

It opens the ticket, the customer does not have to login.

Now you see other assignment? Not the one I assign the ticket to.

Customer can sent comment, but when comment is sent, the admin/assigned group does not get an email about the update by the customer...

Seems to be a bug?

Now you see other assignment? Not the one I assign the ticket to.

I am not sure what you mean.

Customer can sent comment, but when comment is sent, the admin/assigned group does not get an email about the update by the customer...

The users in the group the Ticket is assigned to will get an email when the Customer updates the Ticket. Make sure that the users have a valid email and that email is configured and working properly in your DotNetNuke Host settings.