To send the F4 key to another process you will have to activate that process
http://bytes.com/groups/net-c/230693-activate-other-process suggests:
- Get Process class instance returned by Process.Start
 
- Query Process.MainWindowHandle
 
- Call unmanaged Win32 API function "ShowWindow" or "SwitchToThisWindow"
 
You may then be able to use System.Windows.Forms.SendKeys.Send("{F4}") as Reed suggested to send the keystrokes to this process
EDIT:
The code example below runs notepad and sends "ABC" to it:
using System;
using System.Diagnostics;
using System.Runtime.InteropServices;
using System.Windows.Forms;
namespace TextSendKeys
{
    class Program
    {
        [DllImport("user32.dll")]
        static extern bool ShowWindow(IntPtr hWnd, int nCmdShow);
        static void Main(string[] args)
            {
                Process notepad = new Process();
                notepad.StartInfo.FileName = @"C:\Windows\Notepad.exe";
                notepad.Start();
                // Need to wait for notepad to start
                notepad.WaitForInputIdle();
                IntPtr p = notepad.MainWindowHandle;
                ShowWindow(p, 1);
                SendKeys.SendWait("ABC");
            }
    }
}