Patti,
A new user received an email from our group successfully, but when
she replied to it, she got a mail delivery failure, ...
That doesn't look like a delivery failure notice.
That looks like a notice placed in her reply composition dialog. I think it is really just a warning that Outlook was "concerned" about the sender of the original message (that would be your group's address).
If the blue box at the bottom of the screen shot is the beginning of a quote of the message you sent (to which she's replying) then a mystery to me is why you were listed at the group's address "on behalf of" yourself. Do you use a different email address in that group than here in GMF?
Her message does show up on the website just fine.
...
Since none of the other group members received it, ...
You mean, no other group members reported such a message when they replied? If her message posted to the group's message section then it should have gone also to any members that receive group postings.
Shal
Shal Farley