You don't actually want a group header, because that won't print at any given point on the page (like, say, at the top).  You want to use either the page header or page footer.

The general way to do this is to create a second page header and/or page footer.  Create a global variable.  Create a series of formulas to manipulate the variable, all with WhilePrintingRecords set:
In the Report Header, initialize the variable to 0.
In the Group Header, set the variable to 1.
In the Group Footer, set the variable to 0.
In the "Continued..." page header/footer section, suppress the section if the variable = 0.

In this way, if you get to a page break between a Group Header and a Group Footer, the variable will be 1, which means "Continued..." will be printed.  You will also likely want to tweak the "Keep Together" settings of your sections and groups, to make sure that the pages break in logical places.

For this kind of operation, it's quite simple.

In Report Header, add a formula field that looks like:

WhilePrintingRecords

Global BooleanVar PageTest := False

(This creates the variable PageTest and initializes it to False, meaning we are not in the middle of a group.)

In Group Header:

WhilePrintingRecords

Global BooleanVar PageTest

PageTest = True

In Group Footer:

WhilePrintingRecords

Global BooleanVar PageTest

PageTest = False

In Page Footer:

WhilePrintingRecords

Global BooleanVar PageTest

IF PageTest = True THEN
    "Continued on next page..."
ELSE
    ""

Just add these formulas to the indicated sections, and it should work.  The only formula that will actually show up on the final report will be the last one, which will show the "Continued..." phrase when the page break comes in the middle of the page.