Infidelity Support Group
Any relationship in which one partner engages willfully in sexual relations with another outside of the partnership is considered to have experienced infidelity. This breach of trust is often traumatizing for the faithful partner as well as the relationship, and support is often needed to heal emotionally and to decide whether or not the relationship should continue after...

Good luck
She doesn't want his wife to know, because it's an extra pair of eyes to watch them. Tell his wife because it's the right and moral thing to do. She deserves to know that her husband is chesting on her.
Do you know how long it's been going on?
Have you set on the terms of reconciliation ? Do what you require from her?
I can tell you thst many cheating spouses attend IC AND MC while the affair is still going on. In time, she'll get a burner phone if she knows you keep a track on hers.
Unless she truly believes you will leave her, she has no incentive to change. What repercussions or consequences has she faced, apart from now having to hide it from you?
If she wants to 'fix this', she needs to do the heavy lifting as the one who stepped out of the marriage, otherwise you're wasting your time with false reconciliation. Do you know what you are forgiving? The full extent of it? Tell her to google 'things a wayward spouse should know'
Get tough and be firm. Read 'no more Mr. Nice guy'
Don't let her trample all over your heart.
You are still in emotional ICU, and you're raw, and hurting deeply, and her pushing you away and shutting you out, has intensified that.
You're burdened with the thought of your son, and his group, but there may be another group you can start going to, or you could cool it off in that group for a few weeks, and perhaps this man will leave the group in the meantime. But you can't worry about something that may not happen.
Right now you need to focus your attention on you, so you can start to heal, and salvage your dignity, for yourself and your children. She is in a fog, and only you can know if telling the wife is the right thing to do. Recognize that it could go either way, she may become angry and resentful, or she may want to leave, but I don't believe she can take your children out of their home. I'd ask members who've told and see what they say.
If a reconcilition is going to happen, it will take time, and she has to become transparent and truthful. Right now her feelings for her affair partner are strong, and her tears are real, they're just not for you. But that can change with time.
I wish you well.