C# – Restore a minimized window of another application

I'm adding some code to an app that will launch another app if it isn't already running, or if it is, bring it to the front. This requires a small amount of interop/WinAPI code, which I've gotten examples for from other sites but can't seem to get to work in Win7.

If the window is in some visible state, then the API's SetForegroundWindow method works like a treat (and this would be the main case, as per company policy if the external app is running it should not be minimized). However, if it is minimized (exceptional but important as my app will appear to do nothing in this case), neither this method nor ShowWindow/ShowWindowAsync will actually bring the window back up from the taskbar; all of the methods simply highlight the taskbar button.

Here's the code; most of it works just fine, but the call to ShowWindow() (I've also tried ShowWindowAsync) just never does what I want it to no matter what the command I send is:

[DllImport("user32.dll")]
    private static extern int SetForegroundWindow(IntPtr hWnd);

    private const int SW_SHOWNORMAL = 1;
    private const int SW_SHOWMAXIMIZED = 3;
    private const int SW_RESTORE = 9;

    [DllImport("user32.dll")]
    private static extern bool ShowWindow(IntPtr hWnd, int nCmdShow);

...

//The app is named uniquely enough that it can't be anything else,
//and is not normally launched except by this one.
//so this should normally return zero or one instance
var processes = Process.GetProcessesByName("ExternalApp.exe");

        if (processes.Any()) //a copy is already running
        {
            //I can't currently tell the window's state,
            //so I both restore and activate it
            var handle = processes.First().MainWindowHandle;
            ShowWindow(handle, SW_RESTORE); //GRR!!!
            SetForegroundWindow(handle);
            return true;
        }

        try
        {
            //If a copy is not running, start one.
            Process.Start(@"C:\Program Files (x86)\ExternalApp\ExternalApp.exe");
            return true;
        }
        catch (Exception)
        {
            //fallback for 32-bit OSes
            Process.Start(@"C:\Program Files\ExternalApp\ExternalApp.exe");
            return true;
        }

I've tried SHOWNORMAL (1), SHOWMAXIMIZED (3), RESTORE (9), and a couple other sizing commands, but nothing seems to do the trick. Thoughts?

EDIT: I found an issue with some of the other code I had thought was working. The call to GetProcessesByName() was not finding the process because I was looking for the executable name, which was not the process name. That caused the code I thought was running and failing to actually not execute at all. I thought it was working because the external app will apparently also detect that a copy is already running and try to activate that current instance. I dropped the ".exe" from the process name I search for and now the code executes; however that seems to be a step backwards, as now the taskbar button isn't even highlighted when I call ShowWindow[Async]. So, I now know that neither my app, nor the external app I'm invoking, can change the window state of a different instance programmatically in Win7. What's goin' on here?