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I used to work at a small company of 10 people. I was assigned a few role within the company that would consume up to 5 hours of work per week or month.

We later got a new employee and while chatting with him at the lunch table me and some colleagues figured that he was much more qualified than me to have one of my roles - I pretty much had no idea of what I was doing and was just learning, yet we pretty badly needed that thing done right.

I told him, the new employee, that he could have that role and told him what he needed to know to do it.

Later I spoke with the CEO (my boss) and he just said "alright" when I told him that I had given that role to that colleague. However, things got strange with my relation to the CEO and a few weeks later I was fired.

Did the manager/boss think I'd gone behind his back? That I shouldn't even have recommended the new colleague for that role?

We had always thought about the role as a bit silly since no-one had any expertise in that field back then.

Without knowing the whole sequence of events and internal culture of the organisation, it seems unlikely that you would have been fired for this without it even being mentioned to you by your boss. It sounds like your colleagues were in agreement (and aware of) your assigning the task to the new employee, so it looks like the reassignment of the duty was more of a group decision, rather than just you.

In general, as long as nobody is underperforming with a handover like this (assuming that this role was informal), then there isn't any real issue with passing tasks to another team member. At least not in any times that I have managed.

So the obvious questions to you are, did the new employee perform the role/task well, and did it interfere with their other manager-assigned duties? Even these may not help - if the new employee wasn't coping, I'm sure they would have raised it with you or your manager.

Also, how were you performing in your remaining tasks? While you may be correlating the two incidents (the assignment of the task) with your being fired, perhaps they were unrelated.

We had always thought about the role as a bit silly since no-one had any expertise in that field back then.

This seems to indicate that the role wasn't such a big thing, so adds more likelihood to this not being the reason for your being fired. Unfortunately, the only way you can know is to ask your former employer, who is unlikely to give you a straight answer. But it seems like there might have been other factors than just this that contributed to your being let go.

Did the manager/boss think I'd gone behind his back? That I shouldn't
even have recommended the new colleague for that role?

There's no way to divine what the manager/boss actually thought at the time. I know how I would have felt.

Usually the manager/boss hands out tasks/roles.

I'm guessing this particular role was given to you by your boss. Instead of you deciding on your own to give it away, you should have approached your boss with that suggestion.

Next time, ask first.

However, things got strange with my relation to the CEO and a few
weeks later I was fired.

It's extremely unlikely that you were fired solely for that one act. Hopefully, the CEO mentioned something about why you were fired. If not, you should have asked.