Change Taskbar Icon Using PInvoke

Question

I have an application that launches a windowed process. I have successfully managed to update the window icon as well as the icon displayed when using Alt+Tab using the following:

public class ProcessLauncher {

    ...

    [DllImport("user32.dll")]
    private static extern int SendMessage(IntPtr hwnd, int message, int wParam, IntPtr lParam);

    private const int WM_SETICON = 0x80;
    private const int ICON_SMALL = 0;
    private const int ICON_BIG = 1;

    public void LaunchProcess() {

        var process = new Process();
        var startInfo = new ProcessStartInfo
            {
                FileName = "cmd.exe",
                Arguments = "C:"
            };
        process.StartInfo = startInfo;
        process.Start();

        Thread.Sleep(50);

        var icon = Resources.MyIcon;

        SendMessage(process.MainWindowHandle, WM_SETICON, ICON_BIG, icon.Handle);
        SendMessage(process.MainWindowHandle, WM_SETICON, ICON_SMALL, icon.Handle);
    }

    ...
}

This works just fine, but the taskbar icon for that process remains unchanged. How can I update the taskbar icon to match the icon I've supplied for the window and Alt+Tab?