flake.nix: update during the update script

default.nix

@@ -86,6 +86,23 @@ let
     autoPrUpdate =
       let
         updateScripts = {
+          flake = pkgs.writeShellApplication {
+            name = "update-flake";
+            runtimeInputs = with pkgs; [
+              git
+              nix
+            ];
+            text = ''
+              echo "<details><summary>flake.nix changes</summary>"
+              # Needed because GitHub's rendering of the first body line breaks down otherwise
+              echo ""
+              echo '```'
+              cd "$1"
+              nix flake update 2>&1
+              echo  '```'
+              echo "</details>"
+            '';
+          };
           npins = pkgs.writeShellApplication {
             name = "update-npins";
             runtimeInputs = with pkgs; [ npins ];