unfotunately NO, you CANNOT extend static classes
https://onecompiler.com/csharp/3xvbe7axg
using System;
namespace HelloWorld
{
  public static class console_extensions {
    public static void EXTENSION(this object item) {
      System.Console.WriteLine("HELLO THERE!");
    }
  }
  
    public class Program
    {
        public static void Main(string[] args)
        {
            Console.WriteLine("Hello, World!");
            Console.EXTENSION();
            ((Console)null).EXTENSION();
            Console l = new Console();
            l.EXTENSION();
        }
    }
}
output
Compilation failed: 4 error(s), 0 warnings
HelloWorld.cs(16,12): error CS0117: `System.Console' does not contain a definition for `EXTENSION'
/usr/lib/mono/4.5/mscorlib.dll (Location of the symbol related to previous error)
HelloWorld.cs(17,5): error CS0716: Cannot convert to static type `System.Console'
HelloWorld.cs(18,4): error CS0723: `l': cannot declare variables of static types
/usr/lib/mono/4.5/mscorlib.dll (Location of the symbol related to previous error)
HelloWorld.cs(18,16): error CS0712: Cannot create an instance of the static class `System.Console'
/usr/lib/mono/4.5/mscorlib.dll (Location of the symbol related to previous error)
however you CAN pass null to the extension method
using System;
namespace HelloWorld
{
  public static class static_extensions {
      public static void print(this object item, int data = 0) {
      Console.WriteLine("EXT: I AM A STATIC EXTENSION!");
      Console.WriteLine("EXT: MY ITEM IS: " + item);
      Console.WriteLine("EXT: MY DATA IS: " + data);
      string i;
      if (item == null) {
        i = "null";
      } else {
        i = item.GetType().Name;
      }
      Console.WriteLine("EXT: MY TYPE IS: " + i + "\n");
    }
  }
    public class Program
    {
    
        public static void Main(string[] args)
        {
          // an extension method can be
          //   called directly
          //  (null is an instance)
          static_extensions.print(null);
          // an extension method can also be
          //   called directly with arguments
          //  (null is an instance)
          static_extensions.print(null, 1);
          
          // an extension method can also be
          //   called as part of an instance
          int x = 0; // initialize int
          x.print();
          
          // an extension method can also be
          //   called as part of an instance
          //   and with data
          int x2 = 0; // initialize int
          x2.print(2);
          
          // an extension method can also be
          //   called directly from null
          //   since `null` is an instance
          ((string)null).print();
          
          // an extension method can also be
          //   called directly from null
          //   and with data
          //   since `null` is an instance
          ((string)null).print(4);
        }
    }
}
live example: https://onecompiler.com/csharp/3xvbc8s6w
output:
EXT: I AM A STATIC EXTENSION!
EXT: MY ITEM IS: 
EXT: MY DATA IS: 0
EXT: MY TYPE IS: null
EXT: I AM A STATIC EXTENSION!
EXT: MY ITEM IS: 
EXT: MY DATA IS: 1
EXT: MY TYPE IS: null
EXT: I AM A STATIC EXTENSION!
EXT: MY ITEM IS: 0
EXT: MY DATA IS: 0
EXT: MY TYPE IS: Int32
EXT: I AM A STATIC EXTENSION!
EXT: MY ITEM IS: 0
EXT: MY DATA IS: 2
EXT: MY TYPE IS: Int32
EXT: I AM A STATIC EXTENSION!
EXT: MY ITEM IS: 
EXT: MY DATA IS: 0
EXT: MY TYPE IS: null
EXT: I AM A STATIC EXTENSION!
EXT: MY ITEM IS: 
EXT: MY DATA IS: 4
EXT: MY TYPE IS: null