A brief history:
In the early days of S&W hand ejector revolvers, the company made a .22 rimfire revolver on the tiny M frame, much smaller than the Model of 1896, which became the I frame platform for revolvers chambered for the .32 Smith & Wesson Long cartridge.
In 1910, the company was approached by Philip Bekeart, a San Francisco distributor, who believed that a larger frame rimfire revolver would be a commercial success. After discussion, company officials agreed and the following year produced the first examples of the Bekeart Model, a .22 rimfire revolver built on the .32 frame (I frame) with a 6" barrel and target sights. Only 294 of these guns were produced for Bekeart in 1911. Then production ceased until c. 1914, when the gun was introduced into the S&W catalog and given the name .22/32 Heavy Frame Target.
In January, 1931, the company, at the urging of competitive target shooters, introduced a new, larger .22 rimfire revolver, built on the medium K frame with target sights. This was the first K-22, referred to as the K-22 Outdoorsman. The .22/32 continued in production, but the Heavy Frame Target name was dropped, since it was built on a smaller frame than the Outdoorsman. Henceforth, it would be known as the .22/32 Target revolver.
Then someone in the company had a brilliant idea. Why not use the smaller frame revolver to promote use of a Smith & Wesson rimfire handgun as a camp gun while engaging in fishing and other outdoor activities? So, in 1936, a version of the .22/32 was introduced having a 4" barrel and a round butt. It was to be called the Kit Gun, since it was easily stored in one's fishing or camping kit. The first example carried the serial number 530,003 in the .32 Hand Ejector serial sequence.
Production of the Kit Gun continued until S&W entered full time war production status in the very early 1940s. With the war over, production began again in September, 1950. The postwar Kit Guns were essentially the same but incorporated an internal change to include the improved hammer block safety design that had emerged during the war on the late Victory Model .38 caliber K frame guns. Serial numbers continued to be in the .32 Hand Ejector series.
The prewar Kit Gun had a 5 screw frame, as did the early postwar guns. In 1953, significant engineering changes were made, resulting in the Model of 1953 .22/32 Kit Gun. These included the use of the Improved I frame with its coil mainspring. The visible evidence of this is the absence of the strain screw on the front strap of the grip frame. Moreover, the cylinder stop installation was simplified, eliminating the fifth screw in front of the trigger guard, thus giving the Kit Gun a four screw frame (all four in the sideplate). This newly configured Kit Gun was given a new serial number series that began in 1953 with serial number 101. It should be noted that the .22/32 Target model also received the same modifications in 1953 and shared the serial number sequence with the Kit Gun.
On December 13, 1955, management directed that the top sideplate screw was to be eliminated, making the .22/32 revolvers three screw guns. This change was initiated on the Kit Gun at approximate serial number 11,000. However, as with all S&W revolvers, these changes were phased in. Some frames forged before the change is implemented by the floor supervisor may already be drilled for the fourth screw, and some four screw sideplates may still be in inventory. All parts are used up, since company policy dictates against waste.
Eventually, when S&W decided to assign model numbers to all its handguns, the Kit Gun received the Model 34 assignment, and the .22/32 Target was designated the Model 35. The assignment of model numbers is not related to the different frame screw configurations. In the case of the .22/32, the frame screw alterations came in 1953 and 1955, as related above. The management decision to use model numbers on all S&W handguns was made in July, 1957 - fully two years after the fourth screw was dropped. So there were plenty of non-model-marked Kit Guns with three screw frames. Moreover, so far as we have been able to determine there were no Model 34 revolvers shipped until at least February, 1958 (it may actually have been slightly later than that).
At some date, unknown to me, the Kit Gun was also made available with a 2" barrel, so Kit Guns may be found with either a 2" or 4" barrel.