kernel, image being direct summands

This is embarrassing but I seem to be having trouble with the first isomorphism theorem.

I'm reading stuff about endomorphisms' kernels and images being direct summands of a module. I haven't seen any explicit statements but the authors clearly imply that whether the kernel is a direct summand is indepentent from whether the image is. I don't understand why.

If I have a direct sum of two modules isn't it always true that is isomorphic to Let is clearly an isomorphism, isn't it? I've checked twice and I can't see why it wouldn't be.

But then, for if is a direct summand of say and I get

from the first isomorphism theorem. Which would mean, in particular, that is a direct summand of too. What am I doing wrong?

(I'm sorry if it's not very readable. I hope it is but it's 4.30 am here and I barely see anything.)