Saving the output of a blur shader?

So I have a RawImage on my screen that I want to apply a blur shader to, but then instead of just blurring the final image before it goes to the screen - I want to save that blurred image so that on the next iteration(after it's been drawn to again) I can blur it again.

I have this, but it's CPU code and once the image gets larger than say 200x200, it's too slow to be useful:

Code (CSharp):

1. private void Blur(Texture2D image, int blurSize)
2. {
3.     Color[] blurred = image.GetPixels ();
4.    
5.     // look at every pixel in the blur rectangle
6.     for (int xx = 0; xx < image.width; xx++)
7.     {
8.         for (int yy = 0; yy < image.height; yy++)
9.         {
10.             float avgR = 0, avgG = 0, avgB = 0;
11.             int blurPixelCount = 0;
12.            
13.             // average the color of the red, green and blue for each pixel in the
14.             // blur size while making sure you don't go outside the image bounds
15.             for (int x = xx; (x <= xx + blurSize && x < image.width); x++)
16.             {
17.                 for (int y = yy; (y < yy + blurSize && y < image.height); y++)
18.                 {
19.                     Color pixel = blurred[y*image.width + x];
20.                    
21.                     avgR += pixel.r;
22.                     avgG += pixel.g;
23.                     avgB += pixel.b;
24.                    
25.                     blurPixelCount++;
26.                 }
27.             }
28.            
29.             avgR = avgR / blurPixelCount;
30.             avgG = avgG / blurPixelCount;
31.             avgB = avgB / blurPixelCount;
32.            
33.             // now that we know the average for the blur size, set each pixel to that color
34.             for (int x = xx; x < xx + blurSize && x < image.width; x++)
35.                 for (int y = yy; y < yy + blurSize && y < image.height; y++)
36.                     blurred[y*image.width + x] = new Color(avgR/1.1f-0.002f, avgG/1.1f-0.002f, avgB/1.1f-0.002f);
37.         }
38.     }
39.    
40.     image.SetPixels (blurred);
41. }

What I want to do is this same effect, but with a shader instead. Is there a way to save the final image back into something like _MainTex?

 

Alternatively, and probably more in line with what I want, can I modify a texture from within a shader, so that I can access(and draw to) that texture later.