Q:

How to SSH to and connect back to original network?

Once I’ve connected to my Linux Machine(B) over the internet from my current PC(A), is there a way to run software on that machine(B) on the local one(A) so that it uses the original network. For example to scan the original network using NMap that is installed on the remote machine.
I’ve prepared an beautiful image to illustrate what I mean.

Any links or search terms you could provide would be extremely helpful as I’m not quite sure what to Google for after getting a series of dead ends.

A:

user55325's answer is the correct answer, I will not quibble with that. But, just for the sake of completeness, let me stress you can achieve the same result with OpenSSH alone. 
This is so much easier to do in Linux/Unix that my advice to you is, if your machine A is a Windows PC, to install a Linux Virtual Machine onto your Windows PC, and then follow the instructions that I am about to suggest for setting up a connection between two Unices. Any hypervisor will do, but generally VirtualBox is quite user-friendly. 
Google is full of guides about how to do this, I just searched for ssh vpn. The first guide is Ubuntu-specific; as a matter of taste, being a dedicated Debian/Arch Linux user, I would suggest the Arch-Linux way (the Arch Linux documentation, and especially its wiki, is one of the best around, possibly rivaled only by OpenBSD). Lastly, for a more distro-agnostic introduction, I would suggest Bohdi Zazen's excellent intro.
Like I said, the previous answer is excellent, I am adding this for the  sake of completeness. Here and there in the intros mentioned, you will find claims that this way, or user55325's, is better. It is mostly a matter of taste, IMHO. 

