In a large, heavy-bottomed pot, heat oil and kernels, uncovered, over high heat until first pop. Cover and remove from heat for 1 minute. Return to high heat and shake until popping ceases. Sprinkle with salt and shake to distribute evenly. Measure 2 quarts of popcorn, and reserve.
In a saucepan, carefully combine the sugar and water then add the corn syrup and butter. Bring to a boil and cook to soft crack stage. Pour over 2 quarts of popcorn and mix gently but thoroughly with a wooden spoon. Wearing gloves or baggies, mold into balls around popsicle sticks.