There a number of John Kautz's in Union units. Here are the only two Kautzs in Kentucky Units.

John Kautz

Residence was not listed;
Enlisted on 5/21/1861 at Camp O as a Private.
On 6/3/1861 he mustered into "K" Co. KY 1st Infantry
He was Mustered Out on 6/18/1864 at Covington, KY

John D. Kautz

Residence was not listed;
Enlisted on 5/10/1861 at Murfreesboro, TN as a Sergeant.
On 6/5/1861 he mustered into "D" Co. KY 1st Infantry
(date and method of discharge not given)
He was listed as:
* POW 9/10/1863 Graysville, GA (Confined at Macon, GA & Columbia, SC)
Promotions:
* Sergt 4/17/1862
* 1st Sergt 12/10/1862
* 2nd Lieut 7/1/1863 (Estimated date)
Sources used by Historical Data Systems, Inc.:
- Report of the Adjutant General of the State of Kentucky
- Research by Jack Lundquist
(c) Historical Data Systems, Inc.

Why are you posting under the 1st Kentucky? Do you have information that this was his unit?

George Martin