Combine sifted flour and baking powder in a bowl and set aside. Using a mixer, cream the butter and sugar. Add eggs 1 at a time; mix well after each addition. Next add pureed strawberries and extract and combine well. Gradually add the flour (and baking powder) and the milk alternating between them until completely mixed. Then add lemon zest and juice and mix well.
Lightly grease and flour a 13 X 9 pan or (2) 8 round pans. Pour batter into pans and bake at 350 degrees for 25 - 30 min or until a toothpick inserted in the middle comes out clean. I did (2) 8 pans and it was done perfectly right at 25 minute Oven times may vary.
For the icing : using a mixer, cream the cream cheese and butter until nice and smooth. Add the extract and pureed strawberries. Gradually add the powder sugar until it has the consistency and sweetness you desire. You can alter this to your liking. If you make it a bit too sweet, add just a little bit of lemon juice. This will help balance out the sweetness.