In my experience, rebuilding trust in an organization is achieved when there is both admission of wrong and forgiveness.
Forgiveness is really important. The person who was wronged or betrayed needs to be willing to forgive the betrayer. It’s healing for that person to put down the grudge, to lay down the burden of anger and hurt feelings. However, the entire burden for healing the broken trust does not lay on the wronged person’s shoulders. Both sides need to do some work.
It’s quite difficult to forgive if the person who broke trust doesn’t acknowledge that they were in the wrong, that they understand that it was a breach of trust, that they know why it was a breach, and they accept that the other person/people have a good reason for feeling betrayed. Without such an admission of responsibility AND an apology – a heartfelt one – I doubt there can be healing because there will still be hard feelings on the part of the betrayed one. I’ve seen situations deteriorate over time even if there is a bandaid agreement to work together for a period of time.
Private mediation between the two people most concerned has helped me in the past, with later public acknowledgement to the whole group that the two people now understand each other, that one repents and the other forgives. The rest of the group may then need to voice their feelings about the aftermath of the conflict. Obviously, that has to be mediated, and time-limited.
Having some kind of common understanding of each other and each others’ values can help heal a breach and reduce the possibility of future betrayals. I’ve used the Myers Briggs and DISC instruments as non-judgemental mechanisms to get information out and increase mutual understanding. Myers Briggs is my preferred tool because there is so much great support information for it, and the interpretations are more nuanced.