Most important mistake people make: ALWAYS ask Mary if she’s willing to get the intro. Honestly you’d be surprised by the answers. More often than not: she doesn’t think that would make so much sense / she’s very busy / she doesn’t need another email thread to manage.
Sending email intros takes time. Especially for the person that’s making the introduction, you need to write out clearly and the objective of the mail.
But sometimes, people might forget that the person that you like to introduce might not be interested. A good practice is to send a separate mail and asked first, it works for me. I hope it works for you too.