I'm not related, but did some looking based on your query.I think the name you are seeking in Martinek, Martineck, Martinick, etc.I also believe that Rose M. is probably Josephine, Jossie, Rosie, or the like.
In the 1880 Prairie du Chien census, she is listed as Josephine Murtinick, age 1, daughter of Joseph and Katherine Murtinick.I speculate that the Murtinick/Martinek's stayed in Crawford Co., WI until the late 1890's.
Below is who I believe to be Rose/Josephine's brother.
Per the Crawford Co., WI marriage records:
John "Martinek" married Josephine Fuka, 2 Jan 1893, Prairie du Chien.John was recorded as the son of Joe and Kate (Curat) Martinek.Recorded in State Vol. 3, No. 0473.
John F. Martinek/Martineck married Sarah Beesucker, 5 Oct 1904, Prairie du Chien.John was recorded as the son of "John" and Catherine (Unknown) Martinek.Recorded in State Vol. 3, No. 2430.
This would imply that Katherine/Catherine/Kate's maiden name was something like Curat.
Back to the Fern's.
1895 State Census, Dubuque 4th Ward, Julian Twp., Dubuque Co., IA, 1 Jan 1895, Pg. 179, Household 291:
Benjamin Fern, 66, IRE, Stone Mason, Methodist
Annie E., 67, VA, Keeps House, Methodist
Nancy J., 33, Dubuque Co., IA, Dressmaker, Methodist
Jessie, 25, Dubuque Co., IA, Moulder, Congregational (the religious rebel, I guess)
Bert, 17, Dubuque Co., IA, Carpenter, Methodist
Arthur, 6, Dubuque Co., IA, At School, Methodist
1900 Census, Dubuque 4th Ward, Julian Twp., Dubuque, IA, 6 Jun 1900, Household 140, Atlantic Ave:
Benjamin Fern, Aug 1827, IRE, ENG, ENG, Md. 47 years Stone Mason, Immigrated 1829
Anne E., Nov 1831, VA, VA, VA, 10 children, 8 living
Nancy F./J., Sep 1881, IA, IRE, VA, Dressmaker
1900 Census, Dubuque City, 4th Ward, Dubuque, IA, 6 Jun 1900, Household 144 (3 doors away), 1679 Auburn Ave:
Joseph Martineck, Feb 1833, BOH, BOH, BOH, Day Laborer, Md. 37 years, Immigrated in 1870
Katie, Dec 1843, BOH, BOH, BOH, 10 children, 7 living
Frank, Nov 1874, WI, BOH, BOH, Grocers Clerk
James, Aug 1875, WI, BOH, BOH, ?? Railroad
Peter, Sep 1877, WI, BOH, BOH, Bridge Carpenter
Joseph, Sep 1884, WI, BOH, BOH, Cooper
Mary, Aug 1886, WI, BOH, BOH, At School
Mamie, Dec 1893, WI, BOH, BOH, Granddaughter, At School
This would appear to be Burt and Josephine's parents living close by in 1900 Dubuque.
1900 Census, Dubuque 4th Ward, Julian Twp., Dubuque, IA, 7 Jun 1900, Household 164, 232 Algona Ave:
Burt E. Fern, Sep 1876, IA, IRE, ENG, Md. 2 yrs, Carpenter
Rosie/Josie M., Mar 1880, WI, BOH, BOH
Roy E., Jan 1899 IA, IA, WI
Although the wife's name looks like Rosie M., the writing looks like it was changed to Josie M. (Josephine).
1910 Census, Dubuque City, 4th Ward, Julian Twp., Dubuque, IA, Pg. 74, 1679 Auburn Ave, Household 78:
Joseph F. Martinek, 77, Austria BOH, Aust BOH, Aust BOH, Railroad, Immigrated in 1871
Katherine, 68, Aust BOH, Aust BOH, Aust BOH, Md. 42 yrs, 10 children 7 living
Peter P., 31, WI, Aust BOH, Aust, BOH, Railroad Bridge Builder
Joseph D., 28, WI, Aust BOH, Aust BOH, Aust BOH, Railroad Engineer
Per the 1909 Dubuque City Directory, Pg. 677, Joseph F. Martinek lived at 1679 Auburn Ave.
1910 Census, Clinton Twp., Custer Co., OK, 12 Apr 1910, Pg. 128, Household 49:
Bert W. Fern, 38, IA, ENG, SCOT, Home Carpenter
Josie A., 39, WI, Aust BOH, Aust, BOH, Md. 12 years, 5 children, 5 living
Ray E., 11, IA, IA, WI
Mariette, 9, IA, IA, WI
Harold, 8, IA, IA, WI
Marie E., 2, IA, IA, WI
Keith, 4/12, IA, IA, WI
This seems again to be Josephine/Josie.
1920 Census, Tonkawa Twp., Caddo Co., OK, Pg. 242, 23 Sep 1920, Household 161:
Bert W. Fern, 43, IA, IRE, WV, Farmer
Josie F., 40, WI, BOH, BOH
Roy Edward, 21, IA, IA, WI
May Etta, 18, IA, IA, WI
Harold, 17, IA, IA, WI
Marie, 12, IA, IA, WI
Keith, 9, OK, IA, WI
Kenneth, 8, OK, IA, WI
Myrtle, 5, OK, IA, WI
Bert Jr., 1, OK, IA, WI
As you have them in Council Grove Twp., Oklahoma Co., OK in 1930, Bert's wife seems again to be Josephine A.Lots of middle initials, but she seems to be Josephine consistently through the years.I'm sure her parents were Joseph and Katherine/Catherine Martinek or similar last name from the 1880 Crawford Co., Wisconsin census.They seem to have moved to Dubuque Co., IA near the Fern's and the rest was/is history.
Hopefully my ramblings make some sense in your search.