had tattooed

[color=indigo]1) I had tattooed on my hand the letter B.

Can’t this mean three things:

[color=blue]a) The letter B had been tattooed there. One could imagine a context where it was tattooed without my wanting it.
When I woke up I noticed that I had tattooed on my hand the letter B. My kidnappers must have done it.
[color=darkblue]
b) I had tattooed the letter B on my hand myself.

[color=red]c) I had had someone else tattoo the letter B on my hand.

I am pretty sure it can mean ‘a’ and ‘b’, but I am not sure it could mean ‘c’.

Gratefully,
Navi.

Yes, it could, though why would you use that structure over the far more straightforward, “I had a letter B tattooed on my hand” (which could also mean all those things)?