quotients of quotient groups

Suppose K is a normal subgroup of G, and f : G -> G/K

is the natural map. If N is a normal subgroup of G,

then f(N) is a normal subgroup of G/K, so we can form

the quotient (G/K)/f(N)

But then what is the identity element of (G/K)/f(N) ?

Isn't is K.f(N), since K is the identity element of G/K ?

If H : G/K -> (G/K)/f(N) is the natural map on G/K,

how do you compute ker (Hf) ?
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