You could make one php file (not a page template) for the code of that group.
Then you can include that file on the pages so you only have to edit one file to indirectly edit all the files that included that file.
You can do this with the get_template_part function
You create a php file with the name of the group or something to remember it by like normalpage-code.php. you can place it directly in your (child) theme or create another directory and place it there.
then in your page templates you can do:
<?php get_template_part( 'normalpage','code' ); ?>
or if you have it in another directory in your theme:
<?php get_template_part( '/anotherdirectory/normalpage','code' ); ?>
You could also use the php include function combined with locate_template to pass variables from the page template to the code for the group.
<?php include( locate_template( '/anotherdirectory/normalpage-code.php' ); ?>