I'm not a Betty Grable expert, however based on authentic ones that I have seen these both look pretty good to me. Especially if you look at this thread here which has an authentic Grable:
And it is very likely they did send these to the troops but I don't know about this one for sure. I know she did volunteer a lot of time, money, and wrote letters to soldiers during the war.
Permalink Reply by pjc on February 4, 2016 at 4:26am
These were sent to a fan in the UK, late 1930s early 40s from the studio. He used to write and asked for photos. I was given his collection a few weeks ago. He passed away in 1989.