UPDATE: It will not work correctly for out of bound bounding boxes after scaling.
Code:
Use the scale_factor to control the new rectangle size. Also a different formula can be used for M. Using abs may not be necessary.
import face_recognition
import cv2
img = face_recognition.load_image_file("test.png")
img_rgb = cv2.cvtColor(img, cv2.COLOR_BGR2RGB)
img_rgb_copy = img_rgb.copy()
## Define scale factor and window size
scale_factor = 1.1
sz1 = img_rgb.shape[1] * 2
sz2 = img_rgb.shape[0] * 2
face_locations = face_recognition.face_locations(img_rgb)
for top, right, bottom, left in face_locations:
# Draw a box around the face
#cv2.rectangle(img, (left, top), (right, bottom), (0, 0, 255), 2)
crop_img = img_rgb[top:bottom, left:right]
#cv2.imwrite('test_crop.png', crop_img)
## Calculate center points and rectangle side length
width = right - left
height = bottom - top
cX = left + width // 2
cY = top + height // 2
M = (abs(width) + abs(height)) / 2
## Get the resized rectangle points
newLeft = max(0, int(cX - scale_factor * M))
newTop = max(0, int(cY - scale_factor * M))
newRight = min(img_rgb.shape[1], int(cX + scale_factor * M))
newBottom = min(img_rgb.shape[0], int(cY + scale_factor * M))
## Draw the circle and bounding boxes
cv2.circle(img_rgb_copy, (cX, cY), radius=0, color=(0, 0, 255), thickness=2)
cv2.rectangle(img_rgb_copy, (left, top), (right, bottom), (0, 0, 255), 2)
cv2.rectangle(img_rgb_copy, (newLeft, newTop), (newRight, newBottom), (255, 0, 0), 2)
## Show the original image in window resized to double
cv2.namedWindow('image', cv2.WINDOW_NORMAL)
cv2.resizeWindow('image', sz1, sz2)
cv2.imshow("image", img_rgb_copy)
cv2.waitKey(0)
cv2.destroyAllWindows()
Image:

Method:
Get the center point (cX, cY) of given region and from that get new left corner of image by subtracting same value from both (cX - M,cY - M). So the right corner will be (cX + M, cY + M). You can use a scale factor such as, M * p instead of M, where p will control how large or small the new region will be.
Center Point:
width = right - left
height = bottom - top
centerX = left + (width / 2)
centerY = top + (height / 2)
M = (abs(width) + abs(height)) / 2
0 <= p < 1, for smaller crop than given in a side
p > 1, for larger crop margin
Also the new crop may be out of bounds in image. To solve it something like, newTop = max(0, newTop), newRight = min(imageWidth, newRight) can be done and similar for others. You can find a demo here,
https://www.desmos.com/calculator/bgn9dobkjt
