On page 45 of Beneath a Scarlet Sky the author, Mark Sullivan, writes:

The Germans began a full-scale invasion of Italy from the north, through Austria and Switzerland, and installed Mussolini in a puppet government called the "Italian Socialist Republic," with its capital the tiny town of Salò on Lake Garda, northeast of Milan.

Is this true? Did Germany really invade Italy through Switzerland? Did the Swiss really relax their famed neutrality to allow this?