Your code can be simplified to this:
$fileName = 'test.php';
$fileContent = file_get_contents($fileName);
preg_match_all('/function(.*?)Action()/', $fileContent, $matches);
$functions = $matches[1];
Result ($functions):
Array
(
    [0] =>  edit
    [1] =>  editGroup
)
Following is your code with some changes...
First, check if anything was found, if so, add that to an array. Here is the working code:
$source = "test.php";
$fh = fopen($source,'r');
$m = array();
while ($line = fgets($fh)) {
    if(preg_match_all('/function(.*?)Action()/', $line, $matches)){
        $m[] = $matches[1][0];
    }
}
Result ($m):
Array
(
    [0] =>  edit
    [1] =>  editGroup
)
Since preg_match_all returns the number of full pattern matches, you can use the return to check if anything was found. If you get a hit, add the wanted value to an array so you can get it later.
You were getting some empty results because not all lines will match ;)
Sidenote: As mentioned, you'll end up with something like string(5) " edit" (notice the white space). I don't know preg, so I can't fix it for you. What I can do is suggest you to change to $functions = array_map('trim', $matches[1]);