He was a resident alien. He came to the US in 1913 and became a citizen in 1944, shortly after one of his sons was killed in Normandy. My grandfather was born in 1920.
And if your mother never applied for her Italian citizenship, she wasn't an Italian citizen...
I realize that you're trying to hang on to any thread here, but there's a wide variety of sources that say you're wrong.
Italian Citizenship from: MOTHER - GRANDFATHER - GREAT GRANDFATHERNothing there about my mother serving in the Italian army.Mother - Grandfather - Great Grandfather: Your maternal grandfather was born in your native country, your maternal great grandfather was an Italian citizen at the time of his birth, you were born after January 1st, 1948, and neither you nor your mother nor your grandfather ever renounced your right to Italian citizenship. If citizenship is acquired by birth in your country and you meet all these conditions, you qualify for Italian citizenship jure sanguinis"link
You're getting unreliable information because you're depending on a fee-for-service website advertisement. It's in their best financial interest to make as many people as possible believe they're eligible for Italian citizenship so they can collect more in service and documentation fees.
Better to get your information from Italian Government websites, like the ones I posted.