In addition to @DerekHolt's observation, it is easily possible for the centralizer of part of a rational representation of a $GL_r(F')$ in $GL_n(F)$ (with $F'$ a finite extension of $F$) to be non-trivial, while the centralizer of the whole image is smaller. Then the conjugate of the image by anything properly in the centralizer of the part will create a copy of $GL_r(F')$ that intersects the original in a proper subgroup, and that subgroup is at least as large as the centralizer, so properly larger than the center...

In addition to @DerekHolt's observation, it is easily possible for the centralizer of part of a rational representation of a $GL_r(F')$ in $GL_n(F)$ (with $F'$ a finite extension of $F$) to be non-trivial, while the centralizer of the whole image is smaller. Then the conjugate of the image by anything outside the centralizer of the part will create a copy of $GL_r(F')$ that intersects the original in a proper subgroup, and that subgroup is at least as large as the centralizer, so properly larger than the center...