Given a ring map $f \colon A \to B$, and a $B$-module $M$, $M$ has the structure of an $A$-module, and if this module is finitely generated over $A$, the routine pushFwd in this package will compute such an $A$-module. This is also functorial, in that if a map of $B$-modules (both of which are finitely generated over $A$), then pushFwd(RingMap,Matrix) will return the induced map on $A$-modules.
In an algebraic sense, this is really a pull back, but thinking geometrically, the functions here implement the push forward of a module (or sheaf).
This package was originally implemented by Claudiu Raicu, some changes were introduced by Karl Schwede, and later by David Eisenbud and Mike Stillman.
This documentation describes version 0.6 of PushForward.
The source code from which this documentation is derived is in the file PushForward.m2.