Quick questions on Group Theory - Cosets / Normal Group

Hi - This is just to validate my understanding. You may just say yes/no to questions below (Q4 might need some explanation). If there are deeper concepts involved any pointers would we welcome.

G be a group. H is a sub-group.

Let a,b E G

Hx= Right Coset of H in G, with respect to any x E G

yH= Left Coset of H in G, with respect to any y E G

Q1. If Ha = bH, this does not imply aH = bH

Q2. However if H is Normal then above is true

Q3. If Ha = aH does not imply Hb = bH

Q4. If H in not-normal then can this happen Ha = bH (for just some specific a,b and not every a,b)

Reason why I am asking these - We know well that if every right coset is a left coset; H is Normal and vice-versa.

I am wondering if H is not normal, can it still happen that some right coset is equal to some left coset? If yes what else can we deduce?

Coset with respect to 'zero' of the group is trivial so we may exclude that.

Thanks