Tons have people have commented about how your husband may not know he is doing something that is hard for you. I think that's a fair assessment. And if so, why not just ask him about it?
If you said he could be friends with this person, he really may think he is just being friends. And, of course, there could always be something nefarious, but staying silent about is certainly no help. Is there a reason you think his emailing is a secret (instead of just something he never had the chance to tell you)? And if he has a hard time being straightforward, is it helping to let that fester? Either way, you don't know what's up until you've asked. I know that I've had times where I've done something out of my comfort zone, such as talk to a person who is for some reason threatening to my partner even when I've had explicit permission, and I've specifically made an effort to not bring it up without reason. Been honest and forthcoming, sure, but not brought it up without reason. Transparency is wonderful, honesty is ultimate, but sometimes mentioning something like talking to a friend without a social prompt or basis can seem more incriminating or threatening than not. Just ask!