I found out that when my Skype is working then. When I'm trying to reach to web sites they do not logged in automatically but shows me the log-in pages with the fields filled with my correct authentication data.
So what's left for me to do is just to click on the submit button.
In addition I also found out that it asks to log in into a certain web site only once (while the mac book is working). If you close the laptop and opening it again then it asks from you to log-in once again.
It is a little bit annoying to log-in every time that I want to enter a web site.
Maybe someone familiar with this issue and can provide a solution.
Note: this issue doesn't happens when Skype is turned off.
MacBook Pro (15-inch Mid 2010), OS X Mountain Lion (10.8.2)