perldoc $module rather than perldoc $main_module (RT#101917)

lib/Pod/Weaver/Section/Support.pm

@@ -316,7 +316,7 @@ sub weave_section {
 			command => 'head1',
 			content => 'SUPPORT',
 			children => [
-				$self->_add_perldoc( $zilla ),
+				$self->_add_perldoc( $input ),
 				$self->_add_websites( $zilla ),
 				$self->_add_email( $zilla ),
 				$self->_add_irc( $zilla ),
@@ -401,17 +401,13 @@ sub _add_bugs {
 }
 
 sub _add_perldoc {
-	my( $self, $zilla ) = @_;
+	my( $self, $input ) = @_;
 
 	# Do we have anything to do?
 	return () if ! $self->perldoc;
 
-	# Don't use $zilla->name as some dists' name is different from the actual module...
-	# TODO what if user specified $self->all_modules( 1 )? should this use the current filename?
-	my $main_module = $zilla->main_module->name;
-	$main_module =~ s|^lib/||i;
-	$main_module =~ s/\.pm$//;
-	$main_module =~ s|/|::|g;
+        # Get the name for the module, borrow from Pod::Weaver
+	my $module = Pod::Weaver::Section::Name->_get_docname( $input );
 
 	# TODO add language detection as per RT#63726
 
@@ -426,7 +422,7 @@ EOPOD
 
 			} ),
 			Pod::Elemental::Element::Pod5::Verbatim->new( {
-				content => "  perldoc $main_module",
+				content => "  perldoc $module",
 			} ),
 		],
 	} );