Calling class member from .h file
You need to re-think your layout, and your usage of the .h file.

You should never have code in a .h file unless it is either inline or part of a template class.

A .h file should only contain hints about where other things are. Those things should then be in .cpp or .ino files.
For example your IPmonitoring class would be declared in a .h file, and defined (or implemented) in a .cpp file (or .ino file if you prefer). Any place you want to use that class you #include the .h file. In one place you may choose to create an instance of that class. It is that instance that you then reference with extern.
In the Arduino world it is common to create a single instance of the class in the class's .cpp file and extern it in the .h file so everywhere that you include the .h file instantly has access to that single instance of the object.
This is how most libraries are written.
For example:
// IPMonitoring.h

class IPMonitoring {
    public: 
        void doStuff();
};

extern IPMonitoring ipMon;


And:
// IPMonitoring.cpp

#include &lt;IPMonitoring.h&gt;

IPMonitoring ipMon;

void IPMonitoring::doStuff() {
    // Do something
}

And finally:
// mySketch.ino

#include &lt;IPMonitoring.h&gt;

void setup() {
    ipMon.doSomething();
}

void loop() {
}