First time questioner here. I have found myself perplexed by a problem on exact sequences. I have thoroughly scoured the internet for help but my searches have come up empty. Given a long exact sequence of groups
$0 \to A \to B \to C \to D \to E \to F \to 0$
I want to know if i can use this to create a new short exact sequence
$0 \to B/A \to D/C \to F/E \to 0$
My main problem with this is, for example, in my long exact sequence $A \to 0$ in C since the kernel from $B \to C$ is the image of $A \to B$, so in this case I'm not sure that my idea of reducing to the short exact sequence of quotients would be justified. I find it odd that there was little to nothing about this online, which leads me to believe that I may be very wrong in this case. Any help is greatly appreciated.